|  Hyperpolarization read-out through rapidly rotating fields in the zero- and
low-field regime
---|---
|  Laurynas Dagys,∗a and Christian Bengsa
|  An integral part of _para_ -hydrogen induced polarization (PHIP) methods is
the conversion of nuclear singlet order into observable magnetization. In this
study polarisation transfer to a heteronucleus is achieved through a selective
rotation of the proton singlet-triplet states driven by a combination of a
rotating magnetic field and a weak bias field. Surprisingly we find that
efficient polarisation transfer driven by a STORM (Singlet-Triplet
Oscillations through Rotating Magnetic fields) pulse in the presence of
sub-$\mu$T bias fields requires rotation frequencies on the order of several
kHz. The rotation frequencies therefore greatly exceed any of the internal
frequencies of typical zero- to ultralow field experiments. We further show
that the rotational direction of the rotating field is not arbitrary and
greatly influences the final transfer efficiency. Some of these aspects are
demonstrated experimentally by considering hyperpolarised
(1-${}^{13}\mathrm{C}$)fumarate. In addition, we provide numerical simulations
highlighting the resilience of the STORM pulse against disruptive quadrupolar
coupling partners. In contrast to most of the existing methods, the STORM
procedure therefore represents a promising candidate for quadrupolar decoupled
polarisation transfer in PHIP experiments.

##

††footnotetext: School of Chemistry, Highfield Campus, Southampton, United
Kingdom, SO171BJ. E-mail<EMAIL_ADDRESS>
##  1 Introduction

The inherently low sensitivity of Nuclear Magnetic Resonance (NMR) may be
greatly overcome through the use of hyperpolarization methods 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21. At the core of
these methods is the production of nuclear spin order far from thermal
equilibrium that can lead to signal enhancements of many orders of magnitude.

Particular promising techniques are _para_ -hydrogen induced polarisation
(PHIP) methods 1, 2, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21.
Methods of this type utilise molecular hydrogen gas enriched in its _para_
-spin isomer, typically achieved by passing the cooled gas over an iron oxide
catalyst 2, 1.

For the case of hydrogenative-PHIP (considered here) the _para_ -enriched
hydrogen gas (_para_ -hydrogen) is allowed to react with a suitable precursor
molecule. Upon hydrogenation the nuclear singlet order of the hydrogen gas is
carried over to the product molecule. However, the resulting nuclear singlet
order located on the product molecule is NMR silent. Efficient conversion of
nuclear singlet order into observable magnetization is thus an integral part
of the method.

A number of techniques already exist for this purpose, both at high and low
magnetic fields 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21. High field
methods benefit from the usual advantages, spectral separation between hetero-
and homonuclei, strong pulse schemes with error compensation and applicability
to a broad class of molecular systems 22, 23, 15, 21, 11, 24. However, these
benefits often come at a price. Some technical challenges may arise due to
additional relaxation phenomena and coherent leakage, which may lead to
significant polarization losses 17, 25, 26.

Some of these issues may be circumvented at low magnetic fields 10, 13, 14,
18, 17, 19. This has been utilised to produce large quantities of chemically
pure and hyperpolarized (1-${}^{13}\mathrm{C}$)fumarate, for example 19. The
reaction was carried out inside a pressurised metal reactor, which itself was
placed inside a magnetic shield. The polarization transfer was performed by
sweeping the magnetic field in the sub-microtesla regime. Such a setup would
be impossible at high magnetic fields as the reaction vessel is incompatible
with pulsed radio-frequency methods.

We have recently demonstrated that efficient polarisation transfer may also be
performed in the presence of a weak static magnetic field superimposed with a
weak oscillating low field (WOLF) along the same direction 20. A magnetic
field geometry with the oscillating field applied along the same direction as
the main magnetic field is unusual for NMR, indeed if the oscillating field is
applied in the conventional transverse plane the WOLF pulse becomes
ineffective.

Generally speaking, the description of oscillating fields in the low magnetic
field regime is complicated. In contrast to high field experiments both the
resonant part and the counter-rotating part of the linearly polarised field
have to be considered 27, 28. However, at low magnetic fields it is
technically trivial to generate rotating magnetic fields that are resonant
with the nuclear spin transition frequencies. This way perturbations due to
the counter-rotating components are simply avoided.

In this work we demonstrate that the application of transverse rotating fields
may be exploited for the polarisation transfer step in PHIP experiments. The
application of a suitable STORM (Singlet-Triplet Oscillations through Rotating
Magnetic fields) pulse enables polarisation transfer from the singlet pair to
a heteronucleus leading to substantially enhanced NMR signals. We validate
some of these concepts experimentally by generating hyperpolarized
(1-${}^{13}\mathrm{C}$)fumarate, and explore the STORM condition as a function
of the bias field, rotation frequency and sense of rotation.

We find that driving spin transitions with a rotating magnetic field in low
magnetic fields requires unusually high rotation frequencies, sometimes
several kHz. This is in contrast to typical zero-to-ultralow field experiments
which involve frequencies on the order of several Hz at most 29, 30, 31. The
observed polarization levels are comparable to other low-field techniques 18,
17, 19, 20. However, we provide numerical evidence that the STORM method
greatly outperforms several existing techniques in the presence of fast
relaxing quadrupolar nuclei, such as deuterium for example. The STORM method
might therefore represent a simple solution to quadrupolar decoupled
polarisation transfer at low magnetic fields 32, 33.

##  2 Theory

Consider an ensemble of nuclear three-spin-1/2 systems consisting of two
nuclei of isotopic type $I$ and a third nucleus of isotopic type $S$. The
nuclei are characterised by the magnetogyric ratio’s $\gamma_{I}$ and
$\gamma_{S}$, respectively. For an isotropic solution, the nuclei mutually
interact by scalar spin-spin coupling terms

$\displaystyle H_{J}=H_{II}+H_{IS},$ (1)

where the Hamiltonian $H_{II}$ describes the homonuclear couplings

$\displaystyle H_{II}=2\pi J_{12}{\bf I}_{1}\cdot{\bf I}_{2}$ (2)

and the Hamiltonian $H_{IS}$ describes the heteronuclear couplings

$\displaystyle H_{IS}=2\pi J_{13}{\bf I}_{1}\cdot{\bf S}+2\pi J_{23}{\bf
I}_{2}\cdot{\bf S}.$ (3)

For the remainder of the discussion we assume the coupling constants $J_{13}$
and $J_{23}$ to be different ($J_{13}\neq J_{23}$) and a positive homonuclear
coupling constant $J_{12}$. The heteronuclear $J$ coupling Hamiltonian may be
split in its symmetric and anti-symmetric part

$\displaystyle H_{IS}=H^{\Sigma}_{IS}+H^{\Delta}_{IS},$ (4)

with

$\displaystyle H^{\Sigma}_{IS}=\pi(J_{13}+J_{23})({\bf I}_{1}+{\bf
I}_{2})\cdot{\bf S},$ (5) $\displaystyle
H^{\Delta}_{IS}=\pi(J_{13}-J_{23})({\bf I}_{1}-{\bf I}_{2})\cdot{\bf S}.$

Fig. 1:  Eigenvalues and eigenstates of the effective field Hamiltonian
(equation 16) in the near-equivalence limit ($|J_{13}-J_{23}|\ll|J_{12}|$,
with $J_{12}$ referring to the homonuclear $J$-coupling) and referring
$J_{13}$, $J_{23}$ to the out-of-pair couplings. The circles represent the
population distribution for a fully populated singlet state between the two
$I$-spins. The effective nutation frequency $\omega_{\mathrm{eff}}^{X}$ is
given by equation 13 where $X$ refers to either spin I or S. Primes indicate
alignment of the nuclear spins states along the effective field.

The nuclear spin ensemble may further be manipulated by the application of
external magnetic fields. The magnetic field Hamiltonian is constructed by
coupling the spin angular momenta to the external magnetic field taking their
respective magnetogyric ratio’s into account

$\displaystyle H_{M}(t)=-\gamma_{I}{\bf B}(t)\cdot({\bf I}_{1}+{\bf
I}_{2})-\gamma_{S}{\bf B}(t)\cdot{\bf S}.$ (6)

The total spin Hamiltonian is then given by

$\displaystyle H(t)=H_{J}+H_{M}(t).$ (7)

###  2.1 Rotating field Hamiltonian

Fig. 2:  Eigenvalues and eigenstates of the interaction frame Hamiltonian
(equation 15) as a function of the rotating magnetic field frequency
$\omega_{\rm rot}$ at zero-field. The states
$|S_{0}\beta^{{}^{\prime}}\rangle$ and $|T_{-1}\alpha^{{}^{\prime}}\rangle$
undergo a Level-Anti Crossing (LAC) highlighted by the inset. Note that
$\omega_{1}^{I}/\omega_{1}^{S}\approx 4$ for $I=^{1}$H and $S=^{13}$C.

Consider now the application of a time-dependent rotating magnetic field in
the presence of a weak bias field along the laboratory frame $z$-axis. The
$z$-bias Hamiltonian is given by

$\displaystyle H_{\rm bias}$ $\displaystyle=-\gamma_{I}B_{\rm
bias}(I_{1z}+I_{2z})-\gamma_{S}B_{\rm bias}S_{z}$ (8)
$\displaystyle=\omega^{I}_{0}(I_{1z}+I_{2z})+\omega^{S}_{0}S_{z},$

whereas the rotating magnetic field Hamiltonian is given by

$\displaystyle H_{\rm rot}(t)=$ $\displaystyle B_{\rm rot}\cos(\omega_{\rm
rot}t)(-\gamma_{I}(I_{1x}+I_{2x})-\gamma_{S}S_{x})$ (9) $\displaystyle-$
$\displaystyle B_{\rm rot}\sin(\omega_{\rm
rot}t)(-\gamma_{I}(I_{1y}+I_{2y})-\gamma_{S}S_{y}).$

The total spin Hamiltonian may now be expressed as a combination of scalar-
coupling terms, the bias term and the rotating field contribution

$\displaystyle H(t)=H_{J}+H_{M}(t)=H_{J}+H_{\rm bias}+H_{\rm rot}(t).$ (10)

It turns out to be advantageous to isolate the rotating part of $H(t)$. To
this end we consider an interaction frame transformation rotating all three
spins equally around the laboratory frame $z$-axis. The angular frequency is
chosen to coincide with $\omega_{\rm rot}$

$\displaystyle K_{z}(t)=\exp\\{-{i\mkern
1.0mu}(I_{1z}+I_{2z}+S_{z})\omega_{\rm rot}t\\}.$ (11)

The corresponding interaction frame Hamiltonian $\tilde{H}(t)$ is given by

$\displaystyle\tilde{H}=$ $\displaystyle K_{z}(t)H(t)K_{z}(-t)+{i\mkern
1.0mu}\dot{K}_{z}(t)K_{z}(-t)$ (12) $\displaystyle=$ $\displaystyle
H_{J}+\omega^{I}_{1}(I_{1x}+I_{2x})+\omega^{S}_{1}S_{x}$ $\displaystyle+$
$\displaystyle(\omega^{I}_{0}+\omega_{\rm
rot})(I_{1z}+I_{2z})+(\omega^{S}_{0}+\omega_{\rm rot})S_{z},$

which has the advantage of being time-independent.

###  2.2 Effective field Hamiltonian

Within the interaction frame the spins evolve under a new effective magnetic
field $B_{\rm eff}$. The coupling of the effective field to the $I$ and $S$
spins may be characterised by the effective nutation frequencies $\omega_{\rm
eff}^{X}$

$\displaystyle\omega_{\rm eff}^{X}=\sqrt{(\omega^{X}_{0}+\omega_{\rm
rot})^{2}+(\omega^{X}_{1})^{2}},$ (13)

and the polar angles $\theta_{\rm eff}^{X}$

$\displaystyle\theta_{\rm
eff}^{X}=\operatorname{arctan2}(\omega^{X}_{0}+\omega_{\rm
rot},\omega^{X}_{1}).$ (14)

The polar angles describe the field direction with respect to the laboratory
frame $z$-axis. An alternative representation of $\tilde{H}$ is thus given by

$\displaystyle\tilde{H}=XH_{\rm eff}X^{\dagger}+H^{\Delta}_{IS},$ (15)

where $H_{\rm eff}$ represents the effective field Hamiltonian in the absence
of the anti-symmetric heteronuclear $J$-couplings

$\displaystyle H_{\rm eff}=\omega_{\rm eff}^{I}(I_{1z}+I_{2z})+\omega_{\rm
eff}^{S}S_{z}+H_{II}+H^{\Sigma}_{IS}.$ (16)

The transformation $X$ is defined as a composite rotation of spins $I$ and $S$

$\displaystyle X=R^{12}_{y}(\theta_{\rm eff}^{I})R^{3}_{y}(\theta_{\rm
eff}^{S}).$ (17)

Consider now the set of STZ states aligned along the effective magnetic field
direction

$\displaystyle|T_{m}\mu^{{}^{\prime}}\rangle=X|T_{m}\mu\rangle,$ (18)
$\displaystyle|S_{0}\mu^{{}^{\prime}}\rangle=X|S_{0}\mu\rangle.$

These states are exact eigenstates of the effective field part of the
interaction frame Hamiltonian

$\displaystyle XH_{\rm eff}X^{\dagger}|T_{m}\mu^{{}^{\prime}}\rangle=$
$\displaystyle XH_{\rm eff}X^{\dagger}X|T_{m}\mu\rangle=\lambda
X|T_{m}\mu\rangle=\lambda|T_{m}\mu^{{}^{\prime}}\rangle,$ (19)

and similarly for $|S_{0}\mu^{{}^{\prime}}\rangle$. The rotated STZ states
thus represent the approximate eigenstates of $\tilde{H}$ and the
heteronuclear coupling term $H^{\Delta}_{IS}$ may be considered a
perturbation.

###  2.3 Para-hydrogen Induced Polarisation

For typical PHIP experiments involving $I_{2}S$ systems at sufficiently low
magnetic fields we may approximate the initial state of the spin ensemble by
pure singlet population

$\displaystyle\rho(0)=|S_{0}\rangle\langle
S_{0}|\otimes\frac{1}{2}\mathds{1}_{S},$ (20)

where $\mathds{1}_{S}$ represents the unity operator for spin $S$. Because the
initial state of the ensemble is rotationally invariant we may express
$\rho(0)$ in a straightforward manner in the rotated STZ basis

$\displaystyle\rho(0)=\frac{1}{2}\left(|S_{0}\alpha^{{}^{\prime}}\rangle\langle
S_{0}\alpha^{{}^{\prime}}|+|S_{0}\beta^{{}^{\prime}}\rangle\langle
S_{0}\beta^{{}^{\prime}}|\right).$ (21)

In the absence of $H^{\Delta}_{IS}$ no heteronuclear magnetisation may be
extracted out of the system, however the presence of $H^{\Delta}_{IS}$ causes
coherent mixing within the manifolds
$\\{|S_{0}\beta^{{}^{\prime}}\rangle,|T_{0}\alpha^{{}^{\prime}}\rangle,|T_{-1}\alpha^{{}^{\prime}}\rangle\\}$
and
$\\{|S_{0}\alpha^{{}^{\prime}}\rangle,|T_{0}\beta^{{}^{\prime}}\rangle,|T_{+1}\beta^{{}^{\prime}}\rangle\\}$.

Strictly speaking these two manifolds are not completely isolated from all
other states. But as illustrated in figure 2 mixing of this type will be
efficiently suppressed for our choice of the rotation frequency $\omega_{\rm
rot}$. Taking the first manifold for example, one may show to first order in
perturbation theory that the following inequalities are well satisfied

$\displaystyle|\langle
T_{-1}\alpha^{{}^{\prime}}|\tilde{H}|n\rangle|\ll|\langle
T_{-1}\alpha^{{}^{\prime}}|\tilde{H}|T_{-1}\alpha^{{}^{\prime}}\rangle-\langle
n|\tilde{H}|n\rangle|,$ (22) $\displaystyle|\langle
T_{0}\alpha^{{}^{\prime}}|\tilde{H}|n\rangle|\;\;\ll|\langle
T_{0}\alpha^{{}^{\prime}}|\tilde{H}|T_{0}\alpha^{{}^{\prime}}\rangle-\langle
n|\tilde{H}|n\rangle|,$ $\displaystyle|\langle
S_{0}\beta^{{}^{\prime}}|\tilde{H}|n\rangle|\;\;\ll|\langle
S_{0}\beta^{{}^{\prime}}|\tilde{H}|S_{0}\beta^{{}^{\prime}}\rangle-\langle
n|\tilde{H}|n\rangle|,$

where $|n\rangle$ represents any state outside the manifold.

The energy separation between the $|S_{0}\beta^{{}^{\prime}}\rangle$ and
$|T_{-1}\alpha^{{}^{\prime}}\rangle$ state is given by

$\displaystyle\Delta E=\omega_{\rm eff}^{I}-\omega_{\rm
eff}^{S}+2\pi\left(J_{12}-\frac{J_{13}+J_{23}}{4}\cos(\theta_{\rm
eff}^{I}-\theta_{\rm eff}^{S})\right).$ (23)

As a result, coherent state mixing is maximised by choosing an optimised
rotation frequency $\omega_{\mathrm{STORM}}$ such that

$\displaystyle\Delta E=0\quad{\rm at}\quad\omega_{\rm
rot}=\omega_{\mathrm{STORM}}.$ (24)

We refer to such a scenario as the application of a STORM pulse, which causes
a degeneracy between the $|S_{0}\beta^{{}^{\prime}}\rangle$ and
$|T_{-1}\alpha^{{}^{\prime}}\rangle$ state, and leads to a level anti-crossing
(LAC) if the anti-symmetric heteronuclear $J$-couplings are included (see
inset figure 2) 34, 35.

As shown in the appendix mixing between the $|S_{0}\beta^{{}^{\prime}}\rangle$
and $|T_{-1}\alpha^{{}^{\prime}}\rangle$ state occurs with frequency

$\displaystyle\omega_{ST}^{\rm nut}=$
$\displaystyle\sqrt{2}\pi\cos^{2}(\tfrac{1}{2}(\theta_{\rm
eff}^{I}-\theta_{\rm eff}^{S}))$ (25) $\displaystyle\times$
$\displaystyle(\cos(\xi_{\rm ST})(J_{13}+J_{23})+\sin(\xi_{\rm
ST})(J_{13}-J_{23})),$

where $\xi_{\rm ST}$ represents the mixing angle between the
$|S_{0}\beta^{{}^{\prime}}\rangle$ and $|T_{0}\beta^{{}^{\prime}}\rangle$
state

$\displaystyle\xi_{\rm
ST}=\frac{1}{2}\operatorname{arctan2}(-\frac{J_{12}}{2},\frac{J_{13}-J_{23}}{4}\cos(\theta_{\rm
eff}^{I}-\theta_{\rm eff}^{S})).$ (26)

Application of a rotating magnetic field with $\omega_{\rm
rot}=\omega_{\mathrm{STORM}}$ causes the states
$|S_{0}\beta^{{}^{\prime}}\rangle$ and $|T_{-1}\alpha^{{}^{\prime}}\rangle$ to
approximately follow the dynamics of a two-level system (TLS).

Consider now starting from the density operator in equation 21. The spin-state
populations at time $\tau$ under the TLS approximation are given by 35

$\displaystyle\langle
S_{0}\beta^{{}^{\prime}}|\,\rho(\tau)\,|S_{0}\beta^{{}^{\prime}}\rangle\simeq\tfrac{1}{2}\left(1+\cos(\omega_{ST}^{\mathrm{nut}}\tau)\right),$
$\displaystyle\langle
T_{-1}\alpha^{{}^{\prime}}|\,\rho(\tau)\,|T_{-1}\alpha^{{}^{\prime}}\rangle\simeq\tfrac{1}{2}\left(1-\cos(\omega_{ST}^{\mathrm{nut}}\tau)\right).$
(27)

A complete population inversion between the $|S_{0}\beta^{{}^{\prime}}\rangle$
and $|T_{-1}\alpha^{{}^{\prime}}\rangle$ state may be achieved by applying the
STORM pulse for a duration $\tau^{*}=\pi/\omega_{ST}^{\mathrm{nut}}$. Since
the STORM pulse is only resonant with the
$|S_{0}\beta^{{}^{\prime}}\rangle\leftrightarrow|T_{-1}\alpha^{{}^{\prime}}\rangle$
transition all other states experience negligible evolution. The idealised
density operator after such a STORM pulse is thus given by

$\displaystyle\rho(\tau^{*})\simeq$
$\displaystyle\frac{1}{2}\left(|S_{0}\alpha^{{}^{\prime}}\rangle\langle
S_{0}\alpha^{{}^{\prime}}|+|T_{-1}\alpha^{{}^{\prime}}\rangle\langle
T_{-1}\alpha^{{}^{\prime}}|\right)$ (28) $\displaystyle=$
$\displaystyle\mathds{1}/8+\cos(\theta_{\rm eff}^{S})S_{z}/4+{\rm orth.}\;{\rm
operators},$

which indicates the generation of heteronuclear $S$ spin magnetisation
proportional to $\cos(\theta_{\rm eff}^{S})$. The state of the $S$ spins may
be extracted by tracing over the $I$ spins of the system

$\displaystyle\rho_{S}(\tau^{*})={\rm Tr}_{I}\\{\rho(\tau^{*})\\}=$
$\displaystyle\mathds{1}/2+\cos(\theta_{\rm eff}^{S})S_{z}.$ (29)

As a result, the heteronuclear spins become fully polarised as
$\cos(\theta_{\rm eff}^{S})$ approaches unity. Representing the transfer
amplitude in a slightly more intuitive form:

$\displaystyle\cos(\theta_{\rm eff}^{S})=\frac{\omega^{S}_{0}+\omega_{\rm
rot}}{\sqrt{(\omega^{S}_{1})^{2}+(\omega^{S}_{0}+\omega_{\rm rot})^{2}}},$
(30)

one may see that this condition is met if the sum of the bias $S$ spin Larmor
frequency and angular rotation frequency exceed the $S$ spin nutation
frequency ($|\omega^{S}_{0}+\omega_{\rm rot}|\gg|\omega^{S}_{1}|$).

Although much of the discussion above has focused on the
$\\{|S_{0}\beta^{{}^{\prime}}\rangle,|T_{0}\alpha^{{}^{\prime}}\rangle,|T_{-1}\alpha^{{}^{\prime}}\rangle\\}$
manifold, similar results hold for the
$\\{|S_{0}\alpha^{{}^{\prime}}\rangle,|T_{0}\beta^{{}^{\prime}}\rangle,|T_{+1}\beta^{{}^{\prime}}\rangle\\}$
manifold, only difference being that the relevant energy difference has to be
replaced by

$\displaystyle\Delta E_{(-)}$ $\displaystyle=\omega_{\rm eff}^{I}-\omega_{\rm
eff}^{S}-2\pi\left(J_{12}-\frac{J_{13}+J_{23}}{4}\cos(\theta_{\rm
eff}^{I}-\theta_{\rm eff}^{S})\right),$ (31)

where the $(-)$ indicates that this transition will generally lead to negative
heteronuclear magnetisation.

##  3 Methods

###  3.1 Materials

The precursor solution for fumarate was prepared by dissolving 100 mM disodium
acetylene dicarboxylate, 100 mM sodium sulfite, and 6 mM
$\mathrm{[RuCp^{\ast}(MeCN)_{3}]PF_{6}}$ (CAS number: 99604-67-8) in D2O,
heating to 60${}^{\circ}\mathrm{C}$, and passing through a Millex® 0.22 $\mu$m
PES filter.

_Para_ -hydrogen was produced by passing hydrogen gas over an iron oxide
catalyst packed in a 1/4 inch stainless steel tube cooled by liquid nitrogen
which results in _para_ -enrichment level of 50%.

About 2% of the fumarate molecules contain a naturally-occurring
${}^{13}\mathrm{C}$ nucleus. The two ${}^{1}\mathrm{H}$ nuclei and the
${}^{13}\mathrm{C}$ nucleus form a three-spin-1/2 system of the type discussed
above. The $J$-coupling parameters for the molecular system are consistent
with reference 15, 20.

###  3.2 Equipment

Fig. 3:  Schematic diagram of the experimental setup. (a) Mu-metal shield and
associated components. During the STORM pulse, the Helmholtz coil generates
the bias field $B_{\mathrm{bias}}$ whereas two saddle coils produce the
rotating field $B_{\rm STORM}$. (b) Gas-handling apparatus including a
pressure NMR tube equipped with a capillary for bubbling of the _para_
-enriched $\mathrm{H_{2}}$ gas. MCU - micro-controller unit.

A sketch of the equipment is shown in figure 3. The hydrogen gas is bubbled
through the solution using a 1/16 inch PEEK capillary tube inserted inside a
thin-walled Norell® pressure NMR tube. The Arduino Mega 2560 micro-controller
board was used to actuate the Rheodyne MXP injection valves as well as a power
switch connected to the solenoid coil. The 50 cm long and 15 mm wide coil was
designed to provide a 170 $\mu$T field piercing through the TwinLeaf MS-4 mu-
metal shield. The rotating magnetic field was generated by two 30 cm long
orthogonal saddle coils using a Keysight 33500B waveform generator with two
channels synchronised with phase difference of $\pm 90^{\circ}$. The bias
field was generated by the built-in Helmholtz coil of the Twinleaf shield,
powered by a Keithley 6200 DC current source.

###  3.3 Experimental Procedure

Figure 4 gives an overview of the experimental protocol including the magnetic
field experienced by the sample as a function of time. Each experiment starts
by heating 250$\,\mu\mathrm{L}$ of the sample mixture to $\sim
90{}^{\circ}\mathrm{C}$ in the ambient magnetic field of the laboratory
($\sim$ 110 $\mu$T), followed by insertion into the magnetic shield where a
solenoid coil generates a magnetic field of similar magnitude. _Para_
-enriched hydrogen gas is bubbled through the solution at 6 bar pressure for
30 seconds. The rotating field is generated constantly by the waveform
generator, the amplitude is kept constant ($B_{\rm STORM}=4\,\mu\mathrm{T}$).
The solenoid is switched off by a relay for a period $\tau$ to pre-set the
bias field ($B_{\mathrm{bias}}$) and to mimic a STORM pulse. Afterwards the
sample was removed manually and inserted into the Oxford 400 MHz magnet
equipped with a Bruker Avance Neo spectrometer.

Fig. 4:  Detailed timing diagram for the STORM procedure. (a) A radiofrequency
${}^{13}\mathrm{C}$ pulse is applied in high magnetic field at the end of the
procedure to initiate signal acquisition. (b) Magnetic field profile along the
$z$-axis, showing the ambient laboratory field, the change in field as the
sample is placed in the shield, the bias field $B_{\mathrm{bias}}$ during the
STORM pulse, the removal of the sample from the shield and insertion into the
high-field NMR magnet. (c) Magnetic field generated along the X direction and
(d) Y direction oscillating with frequency $\omega_{\mathrm{STORM}}$ and peak
amplitude $B_{\rm STORM}$. The oscillating fields are generated constantly,
but only resonant for the time period $\tau$.

The ${}^{13}\mathrm{C}$ free-induction decays were initiated by a hard pulse
of 14.7 kHz rf amplitude and recorded with 65 k point density at a spectral
width of 200 ppm. Additional ${}^{1}\mathrm{H}$ decoupling was used for all
experiments. Thermal equilibrium ${}^{13}\mathrm{C}$ spectra were recorded at
room temperature with a recycle delay of 120 s, averaging the signal over 512
transients.

##  4 Results

Figure 5 shows single-transient hyperpolarized ${}^{13}\mathrm{C}$ NMR
spectra, obtained using the STORM procedure as a function of the rotational
direction in the presence of a 0 T bias field (figure 5(a)) and a 2 $\mu$T
bias field (figure 5(b)). In both cases the STORM pulse duration was set to
0.2 s with a peak amplitude of $B_{\rm STORM}=4\leavevmode\nobreak\ \mu$T. The
rotation frequencies, 1150 Hz for (a) and 223 Hz for (b), correspond to the
root of the equation 23. The observed ${}^{13}\mathrm{C}$ polarization levels
clearly depend upon the sense of rotation. At zero field the relevant LAC
conditions for positive and negative ${}^{13}\mathrm{C}$ magnetization are
centred symmetrically around a zero rotation frequency. A simple inversion of
the rotation frequency therefore enables selection of either positive and
negative ${}^{13}\mathrm{C}$ magnetization. In the presence of a non-vanishing
bias field the symmetry with respect to the rotation frequency is broken. This
means that a single frequency can only match one condition and no signal is
observed when the rotation is reversed (see Fig. 5(b)).

Fig. 5:  ${}^{1}\mathrm{H}$-decoupled ${}^{13}\mathrm{C}$ spectra of fumarate
at a field of 9.41 T. Samples subjected to clockwise (green lines) and anti-
clockwise (blue lines) rotating magnetic fields with a peak amplitude of 4
$\mu$T. (a) STORM polarisation transfer at zero-field
($B_{\mathrm{bias}}\simeq 0$ T) and rotation frequency of
$\omega_{\mathrm{STORM}}=1150$ Hz. (b) STORM polarisation transfer inside a
small bias field ($B_{\mathrm{bias}}\simeq 2\;\mu{\rm T}$) and rotation
frequency of $\omega_{\mathrm{STORM}}=223$ Hz. The black trace in (a)
corresponds to a ${}^{13}\mathrm{C}$ NMR spectrum at thermal equilibrium
averaged over 512 transients. Fig. 6:  (a) Hyperpolarized fumarate intensities
as a function of STORM pulse frequency $\omega_{\mathrm{STORM}}$ with a fixed
pulse duration of 200 ms. (b) Hyperpolarized fumarate intensities as a
function of pulse duration $\tau$ with a fixed STORM frequency of 1100 Hz. The
bias field and rotating field are fixed at 0 $\mu$T and 4 $\mu$T,
respectively. Grey lines represent numerical _SpinDynamica_ 36 simulations,
whereas blue lines plot the analytical solution given by equation 2.3. The
intensity scales are normalised to the maximum signal obtained. Fig. 7:  (a)
Hyperpolarized fumarate intensities as a function of STORM pulse frequency
$\omega_{\mathrm{STORM}}$ with a fixed pulse duration of 400 ms. (b)
Hyperpolarized fumarate intensities as a function of pulse duration $\tau$
with a fixed STORM frequency of 222 Hz. The bias field and rotating field are
fixed at 2 $\mu$T and 4 $\mu$T, respectively. Grey lines represent numerical
_SpinDynamica_ 36 simulations, whereas blue lines plot the analytical solution
given by equation 2.3. The intensity scales are normalised to the maximum
signal obtained.

The solid black line in figure 5(a) represents a reference ${}^{13}\mathrm{C}$
spectrum averaged over 512 transients. The spectrum was obtained on the
hydrogenated sample after thermal equilibration. Comparison of these spectra
allows an estimation of the ${}^{13}\mathrm{C}$ polarization levels, which in
this case corresponds to $p^{S}_{z}\simeq 6\%$. These results are comparable
with previous methods under similar experimental conditions. 14, 18, 17, 20
Significant improvements in the polarisation are expected by addressing few
aspects of the setup. Fully enriched _para_ -hydrogen would lead to 3-fold
enhancement, whereas careful optimization of the reaction conditions would
further lead to a better polarization yield14, 18. Some minimal losses could
be also avoided by using a fully automated experimental procedure.

Integrated ${}^{13}\mathrm{C}$ signal amplitudes as a function of the rotation
frequency and the STORM pulse duration $\tau$ at bias field of 0 T and 2
$\mu$T are shown in figures 6 and 7. Each experimental point was obtained from
a separate experiment on a fresh sample. The experimental data has been
normalised to unity to enable a qualitative comparison with numerical
simulations and analytically derived curves based on equation 2.3. The
agreement between both curves and the experimental data is gratifying.

The frequency profiles obtained at different bias fields display a significant
change in their width. At zero-field the full width at half maximum of the
profile was estimated to be $\sim$350 Hz whereas at 2 $\mu$T bias field width
got reduced to $\sim$5 Hz. These matching conditions are much broader than the
0.4 Hz width observed in the profiles using the WOLF method 20. In agreement
with the analytic expression given by equation 25, the polarisation transfer
rate did not vary dramatically with an increase in the bias field. For both
cases, equation 25 correctly predicts the polarisation transfer rate to be
approximately $\sim 2$ Hz, where we have used the J-coupling parameters of
fumarate given in refs 15, 20.

##  5 Conclusions

In this work we have studied the polarization transfer from singlet order to
heteronuclear magnetization in the context of _para_ -hydrogen induced
polarisation. The polarisation transfer is achieved through a combination of a
rotating magnetic field and a small bias field. Despite low values of the
static bias field, the rotation frequency needed to drive the transfer is
strikingly large. However, the principle of the method described here is
rather intuitive and a simple explanation may be given within the framework of
level-anti crossings, at least when the Hamiltonian is expressed within the
interaction frame of the rotating magnetic field. Based on the LAC picture we
were able to establish the resonance conditions for singlet-triplet mixing and
determine the corresponding population transfer rate. In particular, we have
shown that the rotational direction plays an important role in correctly
establishing the resonance condition, and should be chosen carefully in the
experimental context.

There are other methods to convert nuclear singlet order into heteronuclear
polarization, including resonant pulse schemes in high field as well as
magnetic field-cycling at low magnetic fields 10, 12, 11, 16, 17, 18, 13, 14,
19, 20, 21, 15. The STORM method introduced here is conceptually simple and
provides a few advantages over other existing low field methods. The potential
polarization losses caused by additional relaxation effects in high magnetic
fields are entirely avoided with use of low magnetic fields 17, 25, 26.
However, at ultra-low fields quadrupolar nuclei such as 2H or 14N often act as
polarization sinks, and may lead to a significant drops in the polarization
transfer efficiency. The presence of quadrupolar nuclei is expected to be
particularly disrupting to adiabatic field sweep methods, some of which are
routinely utilised in the generation of hyperpolarised
(1-${}^{13}\mathrm{C}$)fumarate 18, 17, 19. In contrast to field sweep
methods, the STORM method allows one to freely select the strength of magnetic
fields as well as the rotation frequency. It is thus conceivable that optimal
conditions for the magnetic field strength and rotation frequency exist at
which the 2H or 14N spins do not interfere with the polarisation transfer
process. We therefore believe that STORM pulses represent promising candidates
for a new class of quadrupolar decoupled polarization transfer methods in the
near future. Applications to other hyperpolarization techniques such as PHIP-
SABRE (Signal Amplification by Reversible Exchange) are also conceivable 9,
11, 12, 25.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

We acknowledge funding received by the Marie Skłodowska-Curie program of the
European Union (grant number 766402), the European Research Council (grant
786707-FunMagResBeacons), and EPSRC-UK (grants EP/P009980/1, EP/P030491/1,
EP/V055593/1). We thank Malcolm H. Levitt for valuable input and support
during preparation of the manuscript.

## Appendix

### STORM nutation frequency

Consider the manifold defined by

$\displaystyle
V_{1}=\\{|S_{0}\beta^{{}^{\prime}}\rangle,|T_{0}\alpha^{{}^{\prime}}\rangle,|T_{-1}\alpha^{{}^{\prime}}\rangle\\},$
(32)

the argument is analogous for the other manifold. The matrix representation of
$\tilde{H}$ restricted to $V_{1}$ manifold is of the following form

$\displaystyle{[}\tilde{H}{]}_{V_{1}}=\left[\begin{array}[]{ccc}-\frac{1}{2}(\omega_{\rm
eff}^{S}+3\pi J_{12})&\frac{\pi}{2}\cos(\theta_{\rm eff}^{I}-\theta_{\rm
eff}^{S})(J_{13}-J_{23})&\blacksquare\\\ \frac{\pi}{2}\cos(\theta_{\rm
eff}^{I}-\theta_{\rm eff}^{S})(J_{13}-J_{23})&-\frac{1}{2}(\omega_{\rm
eff}^{S}-\pi J_{12})&\blacksquare\\\
\blacksquare&\blacksquare&\blacksquare\end{array}\right],$ (33)

where the black squares indicate non-zero, but irrelevant matrix elements.

The energy separation between the $|S_{0}\beta^{{}^{\prime}}\rangle$ and
$|T_{0}\beta^{{}^{\prime}}\rangle$ state equals $|2\pi J_{12}|$ and is not
quite sufficient to perform a TLS approximation. We thus diagonalise the
corresponding subspace making use of the mixing angle

$\displaystyle\xi_{\rm
ST}=\frac{1}{2}\operatorname{arctan2}(-\frac{J_{12}}{2},\frac{J_{13}-J_{23}}{4}\cos(\theta_{\rm
eff}^{I}-\theta_{\rm eff}^{S})).$ (34)

After the diagonalisation process the matrix representation of $\tilde{H}$
takes the form

$\displaystyle{[}\tilde{H}{]}_{V_{1}}=\left[\begin{array}[]{ccc}\blacksquare&0&\frac{1}{2}\omega_{ST}^{\mathrm{nut}}\\\
0&\blacksquare&\blacksquare\\\
\frac{1}{2}\omega_{ST}^{\mathrm{nut}}&\blacksquare&\blacksquare\end{array}\right]\xrightarrow{\rm
TLS\;approximation}\left[\begin{array}[]{cc}\blacksquare&\frac{1}{2}\omega_{ST}^{\mathrm{nut}}\\\
\frac{1}{2}\omega_{ST}^{\mathrm{nut}}&\blacksquare\end{array}\right],$ (35)

where $\omega_{ST}^{\mathrm{nut}}$ is given by equation 25.

## Notes and references

  * Natterer and Bargon 1997 J. Natterer and J. Bargon, _Progress in Nuclear Magnetic Resonance Spectroscopy_ , 1997, 31, 293–315.
  * Bowers and Weitekamp 1987 C. R. Bowers and D. P. Weitekamp, _Journal of the American Chemical Society_ , 1987, 109, 5541–5542.
  * Ardenkjaer-Larsen _et al._ 2003 J. H. Ardenkjaer-Larsen, B. Fridlund, A. Gram, G. Hansson, L. Hansson, M. H. Lerche, R. Servin, M. Thaning and K. Golman, _Proceedings of the National Academy of Sciences of the United States of America_ , 2003, 100, 10158–10163.
  * Maly _et al._ 2008 T. Maly, G. T. Debelouchina, V. S. Bajaj, K.-N. Hu, C.-G. Joo, M. L. Mak–Jurkauskas, J. R. Sirigiri, P. C. A. van der Wel, J. Herzfeld, R. J. Temkin and R. G. Griffin, _The Journal of Chemical Physics_ , 2008, 128, 052211.
  * Walker and Happer 1997 T. G. Walker and W. Happer, _Reviews of Modern Physics_ , 1997, 69, 629–642.
  * Kovtunov _et al._ 2018 K. V. Kovtunov, E. V. Pokochueva, O. G. Salnikov, S. F. Cousin, D. Kurzbach, B. Vuichoud, S. Jannin, E. Y. Chekmenev, B. M. Goodson, D. A. Barskiy and I. V. Koptyug, _Chemistry, an Asian Journal_ , 2018.
  * Emondts _et al._ 2017 M. Emondts, J. F. P. Colell, B. Blümich and P. P. M. Schleker, _Physical Chemistry Chemical Physics_ , 2017, 19, 21933–21937.
  * Stephan _et al._ 2002 M. Stephan, O. Kohlmann, H. G. Niessen, A. Eichhorn and J. Bargon, _Magnetic Resonance in Chemistry_ , 2002, 40, 157–160.
  * Adams _et al._ 2009 R. W. Adams, J. A. Aguilar, K. D. Atkinson, M. J. Cowley, P. I. P. Elliott, S. B. Duckett, G. G. R. Green, I. G. Khazal, J. López-Serrano and D. C. Williamson, _Science_ , 2009, 323, 1708–1711.
  * Jóhannesson _et al._ 2004 H. Jóhannesson, O. Axelsson and M. Karlsson, _Comptes Rendus Physique_ , 2004, 5, 315–324.
  * Theis _et al._ 2014 T. Theis, M. Truong, A. M. Coffey, E. Y. Chekmenev and W. S. Warren, _Journal of Magnetic Resonance_ , 2014, 248, 23–26.
  * Theis _et al._ 2015 T. Theis, M. L. Truong, A. M. Coffey, R. V. Shchepin, K. W. Waddell, F. Shi, B. M. Goodson, W. S. Warren and E. Y. Chekmenev, _Journal of the American Chemical Society_ , 2015, 137, 1404–1407.
  * Cavallari _et al._ 2015 E. Cavallari, C. Carrera, T. Boi, S. Aime and F. Reineri, _The Journal of Physical Chemistry B_ , 2015, 119, 10035–10041.
  * Eills _et al._ 2019 J. Eills, J. W. Blanchard, T. Wu, C. Bengs, J. Hollenbach, D. Budker and M. H. Levitt, _The Journal of Chemical Physics_ , 2019, 150, 174202\.
  * Bengs _et al._ 2020 C. Bengs, L. Dagys and M. H. Levitt, _Journal of Magnetic Resonance_ , 2020, 321, 106850.
  * DeVience _et al._ 2021 S. J. DeVience, M. Greer, S. Mandal and M. S. Rosen, _ChemPhysChem_ , 2021, doi: 10.1002/cphc.202100162.
  * Rodin _et al._ 2021 B. A. Rodin, V. P. Kozinenko, A. S. Kiryutin, A. V. Yurkovskaya, J. Eills and K. L. Ivanov, _Journal of Magnetic Resonance_ , 2021, 327, 106978\.
  * Rodin _et al._ 2021 B. A. Rodin, J. Eills, R. Picazo-Frutos, K. F. Sheberstov, D. Budker and K. L. Ivanov, _Physical Chemistry Chemical Physics_ , 2021, 23, 7125–7134.
  * Knecht _et al._ 2021 S. Knecht, J. W. Blanchard, D. Barskiy, E. Cavallari, L. Dagys, E. V. Dyke, M. Tsukanov, B. Bliemel, K. Münnemann, S. Aime, F. Reineri, M. H. Levitt, G. Buntkowsky, A. Pines, P. Blümler, D. Budker and J. Eills, _Proceedings of the National Academy of Sciences_ , 2021, 118, e2025383118.
  * Dagys _et al._ 2021 L. Dagys, C. Bengs and M. H. Levitt, _The Journal of Chemical Physics_ , 2021, 155, 154201.
  * Dagys _et al._ 2021 L. Dagys, A. P. Jagtap, S. Korchak, S. Mamone, P. Saul, M. H. Levitt and S. Glöggler, _Analyst_ , 2021, 146, 1772–1778.
  * DeVience _et al._ 2013 S. J. DeVience, R. L. Walsworth and M. S. Rosen, _Physical Review Letters_ , 2013, 111, 173002.
  * Pravdivtsev _et al._ 2014 A. N. Pravdivtsev, A. V. Yurkovskaya, N. N. Lukzen, K. L. Ivanov and H.-M. Vieth, _The Journal of Physical Chemistry Letters_ , 2014, 5, 3421–3426.
  * Haake _et al._ 1996 M. Haake, J. Natterer and J. Bargon, _Journal of the American Chemical Society_ , 1996, 118, 8688–8691.
  * Knecht _et al._ 2018 S. Knecht, A. S. Kiryutin, A. V. Yurkovskaya and K. L. Ivanov, _Journal of Magnetic Resonance_ , 2018, 287, 74–81.
  * Knecht _et al._ 2019 S. Knecht, S. Hadjiali, D. A. Barskiy, A. Pines, G. Sauer, A. S. Kiryutin, K. L. Ivanov, A. V. Yurkovskaya and G. Buntkowsky, _The Journal of Physical Chemistry C_ , 2019, 123, 16288–16293.
  * Meriles _et al._ 2004 C. A. Meriles, D. Sakellariou, A. Moulé, M. Goldman, T. F. Budinger and A. Pines, _Journal of Magnetic Resonance_ , 2004, 169, 13–18.
  * Sakellariou _et al._ 2005 D. Sakellariou, C. A. Meriles, R. W. Martin and A. Pines, _Magnetic Resonance Imaging_ , 2005, 23, 295–299.
  * Pileio _et al._ 2009 G. Pileio, M. Carravetta and M. H. Levitt, _Physical Review Letters_ , 2009, 103, 083002.
  * Sjolander _et al._ 2017 T. F. Sjolander, M. C. D. Tayler, A. Kentner, D. Budker and A. Pines, _The Journal of Physical Chemistry Letters_ , 2017, 8, 1512–1516.
  * Sjolander _et al._ 2016 T. F. Sjolander, M. C. D. Tayler, J. P. King, D. Budker and A. Pines, _The Journal of Physical Chemistry A_ , 2016, 120, 4343–4348.
  * Birchall _et al._ 2020 J. R. Birchall, M. S. H. Kabir, O. G. Salnikov, N. V. Chukanov, A. Svyatova, K. V. Kovtunov, I. V. Koptyug, J. G. Gelovani, B. M. Goodson, W. Pham and E. Y. Chekmenev, _Chemical Communications_ , 2020, 56, 9098–9101.
  * Tayler and Gladden 2019 M. C. D. Tayler and L. F. Gladden, _Journal of Magnetic Resonance_ , 2019, 298, 101–106.
  * Rodin and Ivanov 2020 B. A. Rodin and K. L. Ivanov, _Magnetic Resonance_ , 2020, 1, 347–365.
  * Messiah 1962 A. Messiah, _Quantum Mechanics_ , North-Holland, 1962.
  * Bengs and Levitt 2018 C. Bengs and M. H. Levitt, _Magnetic Resonance in Chemistry_ , 2018, 56, 374–414.


# A possibility of Klein Paradox in quaternionic ($3+1$) frame

Geetanjali Pathak and B. C. Chanyal Corresponding author (email:
<EMAIL_ADDRESS>
###### Abstract

In light of the significance of non-commutative quaternionic algebra in modern
physics, the current study proposes the existence of the Klein paradox in the
quaternionic (3+1)-dimensional space-time structure. By introducing
quaternionic wave-function, we rewrite the Klein-Gordon equation in extended
quaternionic form that includes scalar and the vector fields. Because
quaternionic fields are non-commutative, the quaternionic Klein-Gordon
equation provides three separate sets of the probability density and
probability current density of relativistic particles. We explore the
significance of these probability densities by determining the reflection and
transmission coefficients for the quaternionic relativistic step potential.
Furthermore, we also discuss the quaternionic version of the oscillatory,
tunnelling, and Klein zones for the quaternionic step potential. The Klein
paradox occurs only in the Klein zone when the impacted particle’s kinetic
energy is less than $\mathbb{V}_{0}-m_{0}c^{2}$. Therefore, it is emphasized
that for the quaternionic Klein paradox, the quaternionic reflection
coefficient becomes exclusively higher than value one while the quaternionic
transmission coefficient becomes lower than zero.

Keywords: quaternion, wave equation, Klein-Gordon equation, relativistic
particle, Klein paradox.

Department of Physics
G. B. Pant University of Agriculture and Technology, Pantnagar 263 145,
Uttarakhand, India

##  1 Introduction

For the development of the microscopic world in terms of hypercomplex division
algebra, quaternionic quantum mechanics (qQM) has been mathematically
developed as a modified version of quantum theory that uses typical complex
algebra. The quaternions [1] are similar to complex numbers in values, but
their multiplication is non-commutative, giving rise to some additional degree
of freedom. The qQM is formulated using hypercomplex wave functions [2]
evaluated over the quaternionic number field. The quaternions were utilized in
discussions of a novel area of quantum physics [3], where the quaternionic
field characterizes the reducibility of finite dimension. Adler [4] has
researched and compiled information on the quaternionic generalization of
quantum mechanics and quantum field theory. Since the quaternion structure
offers homogeneous space-time in higher-dimensional quantum mechanics and has
the potential to unify all four fundamental interactions, it has replaced the
space-time formalism [5] as the dominant space-time structure in theoretical
physics. Therefore, quaternions are well capable of dealing with higher
dimensions. Furthermore, the quaternions are substantially more compact than
the standard form of Lorentz vectors and tensors for any solution that is
conceivable. Also, quaternionic units are used to define the Pauli spin [2,6]
corresponding to the spin of the particle. As a part of this, the quaternions
were effectively applied in the formulation of a theory for the Dirac
equations, which disclosed various aspects of employing quaternions in
relativistic quantum mechanics [7]. Giardino [8-10] studied the various
applications of quaternionic quantum mechanics by using the concept of real
Hilbert space. The formulation and resolution of the quaternionic form of the
Klein-Gordon equation were the focus of Giardino’s work [11] whereas Seema
[12] provided a pure quaternionic Klein-Gordon equation and quaternionic wave.
The additional parameters of this non-commutative algebra facilitate the
investigation of concealed physical states in complicated circumstances. As
such, the Dirac equation has been derived in quaternionic space [13,14] and
the subject also concerns the formulation of the quaternionic Klein-Gordon
equation for a scalar field [11,12,15]. Not only this, many researchers
[16-28] have used the quaternion algebra for applications in the various
disciplines of physics.

In relativistic qQM, it is very challenging to consider four-dimensional
relativistic quaternionic potential with the application of Klein-Gordon and
Dirac field equations. Initially, Leo et al. [29] provided the key ideas of
quaternionic value potential for solving the quaternionic Dirac equation.
Further, the quantum tunnelling phenomena [30,31] for relativistic fermions
have been studied in the presence of a quaternionic step potential. In this
context, the Klein-paradox [32] is an interesting challenge to discuss in
relativistic qQM, which shows the anomalous reflection of particles off a huge
potential barrier. Many physicists [33-35] worked on the Klein-paradox for
relativistic quantum mechanics. Recently, the role of quaternionic four-spaces
in the fundamental of relativistic quantum mechanics has been explored
[36,37]. However, the Klein-paradox problem can be solved by using
relativistic qQM, which can provide some interesting aspects. Therefore, in
this way, considering the non-commutative nature of quaternions under
multiplication operation and its significance in relativistic qQM, we have
used quaternionic four spaces, i.e. ($3+1$) spaces (where three spaces for
spatial coordinate and one space component for temporal coordinate), to
discuss Klein-Gordon equation for scalar as well as the vector field by using
quaternionic wave function. In the quaternionic field, we propose three
different cases to discuss the probability density and probability current
density of relativistic particles, where one case is precisely matched for a
pure scalar field, the second case corresponds to a pure vector field, and the
third case is applicable to a combined field containing both the scalar and
vector fields. We further define the relativistic step-potential in
quaternionic ($3+1$) spaces to analyze the application of the quaternionic
Klein-Gordon equation. We determine the reflection and transmission
coefficients for a quaternionic plane wavefunction interacting with a
quaternionic relativistic step potential. We offer three separate energy zones
related to particle’s momentum to address different values of the kinetic
energy of particles. As a result, the quaternionic form of the Klein paradox
is established for the proposed Klein zone where the kinetic energy $E$ of
incident particle is less than $\mathbb{V}_{0}-m_{0}c^{2}$. In the foregoing
results, it has been proposed that the Klein paradox occurs when the momentum
direction of particle in step-potential region is opposite to its incident
momentum direction.

##  2 Preliminaries

A quaternionic field is a kind of generalized field made up of four-
dimensional spaces that include respectively, scalar and vector fields. Thus,
a quaternion variable $\mathbb{Q}$ can be expressed as

$\displaystyle\mathbb{Q}\,:\longmapsto\,\,\,\mathbb{Q}(e_{0},e_{j})\,=$
$\displaystyle\,\,e_{0}q_{0}+\sum_{j=1}^{3}e_{j}q_{j}\,\,,\,\,\,\,\forall\,\,q_{0}\in\mathbb{R},\,q_{j}\in\mathbb{R}^{3}\,,$
(1)

where ($e_{0},e_{1},e_{2},e_{3}$) are the quaternionic basis elements in which
$e_{0}$ denotes a scalar unit and $e_{j}$ $\left(\text{for}\,j=1,2,3\right)$
denotes the vector unit. The quaternionic conjugate of variable $\mathbb{Q}$,
defined by $\mathbb{Q}^{*}$, can be written as

$\displaystyle\mathbb{Q^{*}}\,:\longmapsto\,\,\,\mathbb{Q}^{*}(e_{0},-e_{j})\,=$
$\displaystyle\,\,e_{0}q_{0}-\sum_{j=1}^{3}e_{j}q_{j}\,\,.$ (2)

The quaternion conjugation is an involution or its own inverse, it always
gives the original element when applied to an element twice, i.e.,
$(\mathbb{Q}^{*})^{*}=\mathbb{Q}$. The quaternion conjugate of any two
quaternionic variable can also be written as
$(\mathbb{Q}_{1}\mathbb{Q}_{2})^{*}=\mathbb{Q}_{2}^{*}\mathbb{Q}_{1}^{*}$.
However, because the quaternionic multiplication
$(\mathbb{Q}_{1}\mathbb{Q}_{2})$ depends upon the multiplication of their
basis elements, one can write the multiplication relations of quaternion units
as

$\displaystyle e_{0}^{2}\,=$
$\displaystyle\,1,\,\,\,e_{j}^{2}\,=-1,\,\,\,\,e_{0}^{*}\,=\,e_{0},\,\,\,\,e_{j}^{*}\,=\,-e_{j}\,,$
$\displaystyle e_{0}e_{j}\,=$
$\displaystyle\,e_{j}e_{0}\,=\,e_{j},\,\,\,e_{i}e_{j}\,=-\delta_{ij}+\epsilon_{ijk}e_{k},\,\,\,\forall\,(i,j,k=1,2,3),$
(3)

where $\delta_{ij}$ and $\epsilon_{ijk}$ are respectively, the Kronecker delta
and the three index Levi-Civita symbols. Therefore, from Eq.(3), the
multiplication of two quaternions $(\mathbb{Q}_{1}\mathbb{Q}_{2})$ can be
employed as a non commutative quaternionic field as follows:

$\displaystyle\mathbb{Q}_{1}\mathbb{Q}_{2}\,=$
$\displaystyle\,\,e_{0}(q_{0}q^{\prime}_{0}-\boldsymbol{q}.\boldsymbol{q}^{\prime})+\sum_{j=1}^{3}\,e_{j}\left[q_{0}\boldsymbol{q}^{\prime}+q^{\prime}_{0}\boldsymbol{q}+(\boldsymbol{q}\times\boldsymbol{q}^{\prime})_{j}\right]\,.$
(4)

In this situation, the coefficient $e_{0}$ is merely a scalar quantity that
may be stated in the presence of a quaternionic scalar field, but the
coefficient $e_{j}$ can be applied to a pure quaternionic vector field. Thus,
the quaternions show non-commutativity under multiplication operation as
$\mathbb{Q}_{1}\mathbb{Q}_{2}\neq\mathbb{Q}_{2}\mathbb{Q}_{1}$ because of
$(\boldsymbol{q}\times\boldsymbol{q}^{\prime})_{j}\neq(\boldsymbol{q}^{\prime}\times\boldsymbol{q})_{j}$.
The quaternionic multiplication of two quaternions may exhibit a commutative
nature only if the vector products of their vectors $\boldsymbol{q}$ and
$\boldsymbol{q}^{\prime}$ are zero or $\boldsymbol{q}$ is parallel to
$\boldsymbol{q}^{\prime}$. The factor
$\left(\boldsymbol{q}\times\boldsymbol{q}^{\prime}\right)$ plays a significant
role for vector field quantities like spin angular momentum, orbital angular
momentum, etc. Additionally, quaternionic fields are capable of having the
associative property under multiplication, such as
$\mathbb{Q}_{1}\left(\mathbb{Q}_{2}\mathbb{Q}_{3}\right)=\left(\mathbb{Q}_{1}\mathbb{Q}_{2}\right)\mathbb{Q}_{3}.$
The norm of a quaternion is the square root of the product of the quaternion
and its conjugate as

$\displaystyle\mathbb{\left\|Q\right\|\,=}$
$\displaystyle\,\,\sqrt{q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{4}^{3}}\,\,=\,\,\sqrt{\mathbb{Q}\mathbb{Q}^{*}}\,\,=\,\,\sqrt{\mathbb{Q^{*}}\mathbb{Q}}\,,$
(5)

which is always a real number and cannot be negative. The quaternionic norm of
any two variables shows the multiplicative property, which means
$\left\|\mathbb{Q}_{1}\mathbb{Q}_{2}\right\|=\left\|\mathbb{Q}_{2}\right\|\left\|\mathbb{Q}_{1}\right\|$.
In quaternionic division algebra, any nonzero quaternion has an inverse with
regard to the Hamilton product, that is,

$\displaystyle\left[e_{0}q_{0}+\sum_{j=1}^{3}e_{j}q_{j}\right]^{-1}\,\,=$
$\displaystyle\,\,\frac{1}{\sqrt{q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{4}^{3}}}\left[e_{0}q_{0}-\sum_{j=1}^{3}e_{j}q_{j}\right]\,.$
(6)

Moreover, in the quaternionic metric space, the norm makes it possible to
define the distance $\mathbb{D}(\mathbb{Q}_{1},\,\mathbb{Q}_{2})$ between
$\mathbb{Q}_{1}$ and $\mathbb{Q}_{2}$ by the norm of their difference as
$\mathbb{D}(\mathbb{Q}_{1},\,\mathbb{Q}_{2})=\left\|\mathbb{Q}_{1}-\mathbb{Q}_{2}\right\|$.
In terms of the corresponding metric topology, addition and multiplication are
continuous operations.

##  3 Generalized quaternionic wave-function

In modern quantum physics, a quaternionic wave function is a generalized wave
function that characterizes the quantum state mathematically. Thus, the
quaternionic wave-function is somewhat comparable to four space-time wave-
functions that involve four-dimensional non-commutative quaternion algebra.
Now, in order to write the quaternionic wave function, let us start with
quaternionic four$-$space ($\mathbb{X}$) and four$-$linear momentum
($\mathbb{P}$) as

$\displaystyle\mathbb{X}(e_{0},e_{j})\,\,=$
$\displaystyle\,\,e_{0}x_{0}+\sum_{j=1}^{3}e_{j}x_{j}\,\,,\,\,\,\,\forall\,\,x_{0}\in\mathbb{R},\,x_{j}\in\mathbb{R}^{3}\,,$
(7) $\displaystyle\mathbb{P}(e_{0},e_{j})\,\,=$
$\displaystyle\,\,e_{0}p_{0}+\sum_{j=1}^{3}e_{j}p_{j}\,\,,\,\,\,\,\forall\,\,p_{0}\in\mathbb{R},\,p_{j}\in\mathbb{R}^{3}\,,$
(8)

where the coefficient of quaternionic scalar unit ($e_{0}$) can be written as
time component while the vector unit ($e_{j}$) can be written as three-space
components. In quaternionic Euclidean structure ($-,+,+,+$), Eqs. (7) and (8)
can be rewrite as $\mathbb{X}(e_{0},e_{j})=\left(-ict,\,\boldsymbol{x}\right)$
and $\mathbb{P}(e_{0},e_{j})=\left(-iE/c,\,\boldsymbol{p}\right)$; where $c$
is the speed of light. It should be noted that the quaternionic structure has
$(1_{\text{time}}+3_{\text{space}})$ or ($3_{\text{space}}+1_{\text{time}}$)
dimensions, where one dimension is linked to a scalar (time-like) component
and the other three are linked to vector (space-like) components. Therefore,
the quaternionic wave function or a wave state ($\Psi$) can be generalized as,
[12]

$\displaystyle\Psi(e_{0},e_{j})\,=$
$\displaystyle\,\,e_{0}\Psi_{0}\exp\left\\{i\Theta_{0}\right\\}+\sum_{j=1}^{3}e_{j}\Psi_{j}\exp\left\\{i\Theta_{j}\right\\}\,\,,$
(9)

where $(\Psi_{0},\Psi_{j})$ are the scalar and vector parts of quaternionic
state vector while $\left\\{\Theta_{0},\Theta_{j}\right\\}$ are the
corresponding quaternionic phase terms. In the superposition quantum state, we
have both scalar and vector states. For pure quaternionic scalar state we can
write $\Psi(e_{0},0)\,=\,e_{0}\Psi_{0}\exp\left\\{i\Theta_{0}\right\\}$, and
for pure vector state,
$\Psi(0,e_{j})\,=\,\sum_{j=1}^{3}e_{j}\Psi_{j}\exp\left\\{i\Theta_{j}\right\\}$.
The quaternionic phase ($\Theta$) can be employed by the product of conjugate
four-momentum and four-space as

$\displaystyle\Theta(e_{0},e_{j})\,\,:\longmapsto\frac{1}{\hbar}\left(\mathbb{P}^{*}\mathbb{X}\right)$
$\displaystyle\,=\,\,\frac{1}{\hbar}\left[\left(e_{0}p_{0}-\sum_{j=1}^{3}e_{j}p_{j}\right)\left(e_{0}x_{0}+\sum_{j=1}^{3}e_{j}x_{j}\right)\right]$
$\displaystyle=\,\,e_{0}\Theta_{0}+\sum_{j=1}^{3}e_{j}\Theta_{j}\,,$ (10)

along with quaternionic values

$\displaystyle\Theta_{0}\,=$
$\displaystyle\,\,\frac{1}{\hbar}\left(p_{0}x_{0}+\boldsymbol{p}\cdot\boldsymbol{x}\right)\,=\,\,\frac{1}{\hbar}\left(-Et+\boldsymbol{p}\cdot\boldsymbol{x}\right)\,,$
(11) $\displaystyle\Theta_{j}\,=$
$\displaystyle\,\,\frac{1}{\hbar}\left[x_{0}p_{j}+p_{0}x_{j}+\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right]\,=\,\,\frac{i}{\hbar}\left[ct\boldsymbol{p}_{j}-\left(\frac{E}{c}\right)\boldsymbol{x}_{j}-i\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right]\,.$
(12)

Therefore, from Eq.(9) we gets

$\displaystyle\Psi(e_{0},e_{j})\,=$
$\displaystyle\,\,e_{0}\Psi_{0}\exp\left\\{-\frac{i}{\hbar}\left(Et-\boldsymbol{p}\cdot\boldsymbol{x}\right)\right\\}+\sum_{j=1}^{3}e_{j}\Psi_{j}\exp\left\\{-\frac{1}{\hbar}\left[ct\boldsymbol{p}_{j}-\left(\frac{E}{c}\right)\boldsymbol{x}_{j}-i\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right]\right\\}\,\,.$
(13)

Eq.(13) represents a generalized wave state of a particle in terms of
quaternionic basis. Although the real part of the wave state $\Psi(e_{0},0)$
has the conventional significance of representing the dynamics of particle
behavior, while the imaginary part of the quaternionic wave-state
$\Psi(0,e_{j})$ illustrates the damping nature along the $e_{j}-$axes. In the
four-dimensional quaternionic space, the attenuation of the wave state will
always be influenced by the pure quaternionic component. Additionally, it is
straightforward to write the probability of a particle including both
quaternionic components as
$\mathcal{P}\sim\left|\mathcal{P}_{1}-\mathcal{P}_{2}\right|$
$=\left|\Psi_{0}\right|^{2}-\left|\Psi_{j}\right|^{2}\exp\left\\{-2z\right\\}$
where $z=\frac{1}{\hbar c}\left(c^{2}tp_{j}-Ex_{j}\right)$.

##  4 Quaternionic version of Klein-Gordon equation

In order to describe the wave behavior of the particle in a quaternionic
field, we state the Klein-Gordon equation. Thus, let us begin with the
quaternionic eigenvalue equation for the quaternionic momentum of the particle
as

$\displaystyle\mathbb{P}\,\Psi\,=$ $\displaystyle\,\,m_{0}c\,\Psi\,.$ (14)

For the second-order differential equation, we operate complex conjugate of
quaternionic momentum $\mathbb{P}^{*}$ by left to the both side in the Eq.(14)
as

$\displaystyle\mathbb{P}^{*}\left(\mathbb{P}\,\Psi\right)\,=$
$\displaystyle\,\,m_{0}^{2}c^{2}\Psi\,.$ (15)

Using the quaternionic value of $\mathbb{P}^{*},\,\mathbb{P}$ and $\Psi$, we
get

$\displaystyle\mathbb{P}^{*}\left(\mathbb{P}\,\Psi\right)\,=$
$\displaystyle\,\,e_{0}\left\\{\frac{E^{2}}{c^{2}}-\left(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\right)\right\\}\Psi_{0}+\sum_{j=1}^{3}e_{j}\left\\{\frac{E^{2}}{c^{2}}-\left(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\right)\right\\}\Psi_{j}\,\,.$
(16)

So, from Eq.(15) we obtain four relativistic quaternionic equations:

$\displaystyle E^{2}\Psi_{0}\,=$
$\displaystyle\,\,\left(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\right)c^{2}\Psi_{0}+m_{0}^{2}c^{4}\Psi_{0}\,,\,\,\,\,\,\,\,\,\,\,\,\,(\text{coefficient
of }e_{0})$ (17) $\displaystyle E^{2}\Psi_{j}\,=$
$\displaystyle\,\,\left(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\right)c^{2}\Psi_{j}+m_{0}^{2}c^{4}\Psi_{j}\,,\,\,\,\,\,\,\,\,\,\,\,\,(\text{coefficient
of }e_{j})\,\,.$ (18)

Now, to check these equations in quaternionic space-time frame, we put
operators $E=i\hbar\frac{\partial}{\partial t}$ and
$p=-i\hbar\frac{\partial}{\partial x}$, then the generalized quaternionic wave
equations will be written as

$\displaystyle
e_{0}\left(\boxdot-\frac{m_{0}^{2}c^{2}}{\text{$\hbar$}^{2}}\right)\Psi_{0}\,=$
$\displaystyle\,\,0\,,\,\,\,\,\,\,\,(\text{along scalar field)}\,;$ (19)
$\displaystyle
e_{j}\left(\boxdot-\frac{m_{0}^{2}c^{2}}{\text{$\hbar$}^{2}}\right)\Psi_{j}\,=$
$\displaystyle\,\,0\,,\,\,\,\,\,\,\,(\text{along vector field)}\,;$ (20)

where $\boxdot=\left(\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\right)$ represents the d’ Alembert operator. Eqs.(19) and (20)
represent the quaternionic Klein-Gordon equation, respectively for scalar
field corresponding to scalar coefficient $e_{0}$ and vector field
corresponding to coefficient $e_{j}$. It is interesting to note that we may
claim that the generalized quaternionic field can unify relativistic Klein-
Gordon wave equations for the scalar and vector fields in a single frame, such
as

$\displaystyle
e_{0}\left(\boxdot-\kappa^{2}\right)\Psi_{0}+\sum_{j=1}^{3}e_{j}\left(\boxdot-\kappa^{2}\right)\Psi_{j}$
$\displaystyle\,\,=\,0\,,\,\,\,\,\,\,\,\,\,(\kappa=m_{0}c/\hbar)\,.$ (21)

The generalized quaternionic Klein-Gordon equation is applicable if we have a
set of four electric or magnetic potentials that contain both scalar and
vector potentials. Moreover, we may also obtain the group and phase velocities
using the quaternionic wave function (13), which are, respectively,
$v_{g}=c^{2}k/\omega$ and $v_{p}=\pm c.$ As a result, a relation $v_{g}\times
v_{p}=c^{2}$ may be used to express the relationship between group and phase
velocity for the relativistic particles.

##  5 Generalized quaternionic continuity equation and the probability
densities

In this section, we describe how the quaternionic scalar and vector fields
flow in a four-dimensional space-time frame. Thus, to formulate the
generalized continuity equation for a quaternionic relativistic wave
propagation, we may use the quaternionic Klein-Gordon equation given in
Eq.(21) as

$\displaystyle\left(\boxdot-\frac{m^{2}c^{2}}{\text{$\text{$\hbar$}$}^{2}}\right)\Psi\,\equiv\,\,\left(\boxdot-\frac{m^{2}c^{2}}{\text{$\text{$\hbar$}$}^{2}}\right)\left(e_{0}\Psi_{0}+\sum_{j=1}^{3}e_{j}\Psi_{j}\right)\,=$
$\displaystyle\,\,0\,\,,$ (22)

and its quaternionic conjugate

$\displaystyle\left(\text{$\boxdot$}-\frac{m^{2}c^{2}}{\text{$\text{$\hbar$}$}^{2}}\right)\Psi^{*}\,\equiv\,\,\left(\boxdot-\frac{m^{2}c^{2}}{\text{$\text{$\hbar$}$}^{2}}\right)\left(e_{0}\Psi_{0}^{*}-\sum_{j=1}^{3}e_{j}\Psi_{j}^{*}\right)\,=$
$\displaystyle\,\,0\,\,.$ (23)

Now, multiplying the left side of Eq.(22) by quaternionic $\Psi^{*}$ and
Eq.(23) by $\Psi$ and then subtracting both equations gives us the following
simplified version:

$\displaystyle\frac{i\text{$\hbar$}}{2mc^{2}}\frac{\partial}{\partial
t}\left\\{\left[\Psi_{0}\frac{\partial}{\partial
t}\Psi_{0}^{*}-\Psi_{0}^{*}\frac{\partial}{\partial
t}\Psi_{0}\right]+\left[\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)\frac{\partial}{\partial
t}\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)^{*}-\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)^{*}\frac{\partial}{\partial
t}\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)\right]\right\\}$
$\displaystyle+\frac{i\text{\text{$\hbar$}}}{2mc^{2}}\frac{\partial}{\partial
t}\left\\{\left[\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)\frac{\partial}{\partial
t}\Psi_{0}^{*}-\Psi_{0}^{*}\frac{\partial}{\partial
t}\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)\right]+\left[\Psi_{0}\frac{\partial}{\partial
t}\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)^{*}-\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)^{*}\frac{\partial}{\partial
t}\Psi_{0}\right]\right\\}$
$\displaystyle+\frac{i\text{\text{$\hbar$}}}{2m}\boldsymbol{\nabla}\left\\{\left[\Psi_{0}^{*}\boldsymbol{\nabla}\Psi_{0}-\Psi_{0}\boldsymbol{\nabla}\Psi_{0}^{*}\right]+\left[\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)^{*}\boldsymbol{\nabla}\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)-\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)\boldsymbol{\nabla}\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)^{*}\right]\right\\}$
$\displaystyle+\frac{i\text{\text{$\hbar$}}}{2m}\boldsymbol{\nabla}\left\\{\left[\Psi_{0}^{*}\boldsymbol{\nabla}\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)-\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)\boldsymbol{\nabla}\Psi_{0}^{*}\right]+\left[\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)^{*}\boldsymbol{\nabla}\Psi_{0}-\Psi_{0}\boldsymbol{\nabla}\left(\sum_{j=1}^{3}e_{j}\Psi_{j}\right)^{*}\right]\right\\}=0\,.$
(24)

Furthermore, to compare with quaternionic continuity-like equation, we
estimate the quaternionic probability density (qPD) and quaternionic
probability current density (qPCD) from Eq.(24) as follows:

$\displaystyle\rho_{\text{qPD}}(e_{0},e_{j})\,=\,$
$\displaystyle\,\,e_{0}\left[\frac{i\text{\text{$\hbar$}}}{2mc^{2}}\left(\Psi_{0}\frac{\partial}{\partial
t}\Psi_{0}^{*}-\Psi_{0}^{*}\frac{\partial}{\partial
t}\Psi_{0}\right)+\frac{i\text{\text{$\hbar$}}}{2mc^{2}}\left(\sum_{j=1}^{3}\Psi_{j}\frac{\partial}{\partial
t}\Psi_{j}^{*}-\sum_{j=1}^{3}\Psi_{j}^{*}\frac{\partial}{\partial
t}\Psi_{j}\right)\right]$
$\displaystyle\,+\sum_{j=0}^{3}e_{j}\left[\frac{i\text{\text{$\hbar$}}}{2mc^{2}}\left(\Psi_{j}\frac{\partial}{\partial
t}\Psi_{0}^{*}-\Psi_{0}^{*}\frac{\partial}{\partial
t}\Psi_{j}\right)+\frac{i\text{\text{$\hbar$}}}{2mc^{2}}\left(\Psi_{j}^{*}\frac{\partial}{\partial
t}\Psi_{0}-\Psi_{0}\frac{\partial}{\partial t}\Psi_{j}^{*}\right)\right]\,\,,$
(25) $\displaystyle J_{\text{qPCD}}(e_{0},e_{j})\,=$
$\displaystyle\,\,e_{0}\left[\frac{i\text{\text{$\hbar$}}}{2m}\left(\Psi_{0}^{*}\boldsymbol{\nabla}\Psi_{0}-\Psi_{0}\boldsymbol{\nabla}\Psi_{0}^{*}\right)+\frac{i\text{\text{$\hbar$}}}{2m}\left(\sum_{j=1}^{3}\Psi_{j}^{*}\boldsymbol{\nabla}\Psi_{j}-\sum_{j=1}^{3}\Psi_{j}\boldsymbol{\nabla}\Psi_{j}^{*}\right)\right]$
$\displaystyle\,+\sum_{j=0}^{3}e_{j}\left[\frac{i\text{\text{$\hbar$}}}{2m}\left(\Psi_{0}^{*}\boldsymbol{\nabla}\Psi_{j}-\Psi_{j}\boldsymbol{\nabla}\Psi_{0}^{*}\right)+\frac{i\text{\text{$\hbar$}}}{2m}\left(\Psi_{0}\boldsymbol{\nabla}\Psi_{j}^{*}-\Psi_{j}^{*}\boldsymbol{\nabla}\Psi_{0}\right)\right]\,\,.$
(26)

It is noticed that, $\rho_{\text{qPD}}$ and $J_{\text{qPCD}}$ are there self
both the quaternionic probability densities functions along quaternionic basis
$e_{0}$ and $e_{j}.$ To interpret qPD and qPCD, let us take the three cases as

Case 1:
$\displaystyle\rho_{\text{qPD}}=\frac{i\text{\text{$\hbar$}}}{2mc^{2}}\left(\Psi_{0}\frac{\partial}{\partial
t}\Psi_{0}^{*}-\Psi_{0}^{*}\frac{\partial}{\partial
t}\Psi_{0}\right),\,\,\,\,\,\,J_{\text{qPCD}}=\frac{i\text{\text{$\hbar$}}}{2m}\left(\Psi_{0}^{*}\boldsymbol{\nabla}\Psi_{0}-\Psi_{0}\boldsymbol{\nabla}\Psi_{0}^{*}\right),\,\,\,\,\,\,(\text{for
scalar quaternion field)}$ Case 2:
$\displaystyle\rho_{\text{qPD}}=\frac{i\text{\text{$\hbar$}}}{2mc^{2}}\sum_{j=1}^{3}\left(\Psi_{j}\frac{\partial}{\partial
t}\Psi_{j}^{*}-\Psi_{j}^{*}\frac{\partial}{\partial
t}\Psi_{j}\right),\,\,\,\,\,\,J_{\text{qPCD}}=\frac{i\text{\text{$\hbar$}}}{2m}\sum_{j=1}^{3}\left(\Psi_{j}^{*}\boldsymbol{\nabla}\Psi_{j}-\Psi_{j}\boldsymbol{\nabla}\Psi_{j}^{*}\right),\,\,\,\,(\text{for
pure quaternion field)}$ Case 3:
$\displaystyle\begin{cases}\begin{array}[]{cc}\rho_{\text{qPD}}=&\frac{i\text{\text{$\hbar$}}}{2mc^{2}}\sum_{j=1}^{3}\left[\left(\Psi_{j}\frac{\partial}{\partial
t}\Psi_{0}^{*}-\Psi_{0}^{*}\frac{\partial}{\partial
t}\Psi_{j}\right)+\left(\Psi_{j}^{*}\frac{\partial}{\partial
t}\Psi_{0}-\Psi_{0}\frac{\partial}{\partial t}\Psi_{j}^{*}\right)\right]\,,\\\
J_{\text{qPCD}}=&\frac{i\text{\text{$\hbar$}}}{2m}\sum_{j=1}^{3}\left[\left(\Psi_{0}^{*}\nabla\Psi_{j}-\Psi_{j}\nabla\Psi_{0}^{*}\right)+\left(\Psi_{0}\nabla\Psi_{j}^{*}-\Psi_{j}^{*}\nabla\Psi_{0}\right)\right]\,,\end{array}&(\text{for
combined quaternion field)}\end{cases}$

In the above instance, case 1 illustrates the conventional representation of
the probability density and probability current density of a relativistic
particle, but case 2 and case 3 take into account the probability density and
probability current density due to the effect of quaternionic fields. The
probability density and probability current density of a pure quaternionic
field depend solely on the vector fields, whereas for the combined
quaternionic field, they are influenced by the interactions of scalar and
vector fields. A somewhat similar outcome has previously been suggested [12].

##  6 Quaternionic relativistic step-potential and the Klein Paradox

In this section, we will discuss the application of the quaternionic Klein-
Gordon wave to the interaction with a quaternionic step potential. We extend
the idea of Giardino [8-11] where the quaternionic quantum wave equation has
been studied for a particle in a relativistic box. Thus, let us consider a
relativistic quaternionic step-potential
$\mathbb{V}(\mathbb{X}):\rightarrow\mathbb{V}(t,\,x,y,z)$ or
$\mathbb{V}(x,y,z,\,t)$ in four dimensional structure as

$\displaystyle\mathbb{V}(x,y,z,t)\,\,=$
$\displaystyle\,\,\begin{cases}0\,\,,&\,\,\text{for\,\,\,}\mathbb{X}<0\\\
\mathbb{V}_{0}\,\,,&\,\,\text{for\,}\,\,\mathbb{X}\geq 0\,\,\,.\end{cases}$
(27)

Figure 1: Schematic diagram of quaternionic step potential

A relativistic quaternionic Klein-Gordon equation for the step potential of
regions $\mathbb{X}<0$ and $\mathbb{X}\geq 0$ shown in Fig. 1 takes form,

$\displaystyle\left[-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}+\nabla^{2}-\frac{m^{2}c^{2}}{\text{$\hbar$}^{2}}\right]\Psi(e_{0},e_{j})\,=$
$\displaystyle\,\,0\,,\,\,\,\,\,\,\,\,\,\,(\mathbb{X}<0)\,,$ (28)
$\displaystyle\left[\left(\frac{i}{c}\frac{\partial}{\partial
t}-\mathbb{V}_{0}\right)^{2}+\nabla^{2}-\frac{m^{2}c^{2}}{\text{\text{$\hbar$}}^{2}}\right]\Psi(e_{0},e_{j})\,=$
$\displaystyle\,\,0\,,\,\,\,\,\,\,\,\,\,\,(\mathbb{X}\geq 0)\,\,.$ (29)

The quaternionic wave function is now defined as

$\displaystyle\Psi_{\text{$\mathbb{X}$$<0$}}\,\simeq$
$\displaystyle\,\,\Psi_{\mathcal{I}}+\Psi_{\mathcal{R}}\,,\,\,\,\,\,\,\,\,\,\,(\mathbb{X}<0)$
(30)

where $\Psi_{\mathcal{I}}$ is the quaternionic form of incident wave and
$\Psi_{\mathcal{R}}$ is the quaternionic form of reflected wave, which can be
further expressed as,

$\displaystyle\Psi_{\mathcal{I}}(e_{0},e_{j})\,=$
$\displaystyle\,\,\exp\left[-\frac{i}{\text{$\hbar$}}\left(Et-\boldsymbol{p}\cdot\boldsymbol{x}\right)\right]+\sum_{j=1}^{3}e_{j}\exp\left[-\frac{i}{\text{$\hbar$}}\left(-ict\boldsymbol{p}+\frac{iE}{c}\boldsymbol{x}-\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right)\right]\,\,,$
(31)

and

$\displaystyle\Psi_{\mathcal{R}}(e_{0},e_{j})\,=$
$\displaystyle\,\,\mathcal{R}\exp\left[-\frac{i}{\text{\text{$\hbar$}}}\left(Et+\boldsymbol{p}\cdot\boldsymbol{x}\right)\right]+\sum_{j=1}^{3}e_{j}\mathcal{R}\exp\left[-\frac{i}{\text{\text{$\hbar$}}}\left(-ict\boldsymbol{p}-\frac{iE}{\text{$c$}}\boldsymbol{x}+\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right)\right]\,\,,$
(32)

where $\mathcal{R}$ represents the amplitude for the reflected wave. As such,
the quaternionic wave function for $\mathbb{X}\geq 0$ becomes
$\Psi_{\text{$\mathbb{X}\geq 0$}}=\Psi_{\mathcal{T}}$, i.e.,

$\displaystyle\Psi_{\mathcal{T}}(e_{0},e_{j})\,=$
$\displaystyle\,\,\mathcal{T}\exp\left[-\frac{i}{\text{\text{$\hbar$}}}\left(Et-\boldsymbol{p}^{\prime}\cdot\boldsymbol{x}\right)\right]+\sum_{j=1}^{3}e_{j}\mathcal{T}\exp\left[-\frac{i}{\text{\text{$\hbar$}}}\left(-ict\boldsymbol{p}^{\prime}+\frac{iE}{\text{\text{$c$}}}\boldsymbol{x}-\left(\boldsymbol{p}^{\prime}\times\boldsymbol{x}\right)_{j}\right)\right],\,\,\,(\mathbb{X}\geq
0)\,,$ (33)

with transmission amplitude $\mathcal{T}$ and momentum
$\left|\boldsymbol{p^{\prime}}\right|=\,\sqrt{\frac{1}{c^{2}}\left(E-\mathbb{V}_{0}\right)^{2}-m_{0}^{2}c^{2}}$.
Here, it is noticed that the kinetic energy of the transmitted particle will
be lesser than that of the incident kinetic energy ($E$) because of the effect
of step potential. Now, to solve the reflection and transmission coefficients
for the relativistic step potential, we first calculate the reflected and
transmitted amplitude of the quaternionic wave function by applying boundary
conditions. Let us consider the boundary conditions

$\displaystyle\left.\Psi_{\text{$\mathbb{X}$$<0$}}\right|_{\left(0,\,t\right)}\,=$
$\displaystyle\,\,\left.\Psi_{\text{$\mathbb{X}\geq
0$}}\right|_{\left(0,\,t\right)}\,\,;$ (34)
$\displaystyle\left.\dot{\Psi}_{\text{$\mathbb{X}$$<0$}}\right|_{\left(0,\,t\right)}\,=$
$\displaystyle\,\,\left.\dot{\Psi}_{\text{$\mathbb{X}\geq
0$}}\right|_{\left(0,\,t\right)}\,\,,$ (35)

which gives

$\displaystyle 1+\mathcal{R}$
$\displaystyle\,=\,\,\frac{\exp\left[-\frac{i}{\text{\text{$\hbar$}}}Et\right]+\sum_{j=1}^{3}e_{j}\exp\left[-\frac{ct}{\text{$\hbar$}}\boldsymbol{p}^{\prime}_{j}\right]}{\exp\left[-\frac{i}{\text{\text{$\hbar$}}}Et\right]+\sum_{j=1}^{3}e_{j}\exp\left[-\frac{ct}{\text{$\hbar$}}\boldsymbol{p}_{j}\right]}\mathcal{T}\,;$
(36) $\displaystyle 1-\mathcal{R}$
$\displaystyle\,=\,\,\frac{\frac{i}{\text{\text{$\hbar$}}}\boldsymbol{p}^{\prime}\exp\left[-\frac{i}{\text{\text{$\hbar$}}}Et\right]+\sum_{j=1}^{3}\frac{E}{\text{\text{$\hbar$}}c}e_{j}\exp\left[-\frac{ct}{\text{\text{$\hbar$}}}\boldsymbol{p}^{\prime}_{j}\right]}{\frac{i}{\text{\text{$\hbar$}}}\boldsymbol{p}\exp\left[-\frac{i}{\text{\text{$\hbar$}}}Et\right]+\sum_{j=1}^{3}\frac{E}{\text{\text{$\hbar$}}c}e_{j}\exp\left[-\frac{ct}{\text{\text{$\hbar$}}}\boldsymbol{p}_{j}\right]}\mathcal{T}\,,$
(37)

dividing (36) by (37), then simplifying, we get

$\displaystyle|\mathcal{R}|^{2}\,=$
$\displaystyle\mathcal{\,\,R}\mathcal{R}^{*}$ $\displaystyle=$
$\displaystyle\,\frac{\left(\boldsymbol{p}-\boldsymbol{p}^{\prime}\right)^{2}+\left\\{\boldsymbol{p}e^{\left[-\frac{ct}{\text{\text{$\hbar$}}}\boldsymbol{p}^{\prime}\right]}-\boldsymbol{p}^{\prime}e^{\left[-\frac{ct}{\text{\text{$\hbar$}}}\boldsymbol{p}\right]}\right\\}^{2}+\frac{E^{2}}{c^{2}}\left\\{e^{\left[-\frac{ct}{\text{\text{$\hbar$}}}\boldsymbol{p}^{\prime}\right]}-e^{\left[-\frac{ct}{\text{\text{$\hbar$}}}\boldsymbol{p}\right]}\right\\}^{2}}{\left(\boldsymbol{p}+\boldsymbol{p}^{\prime}\right)^{2}+\frac{4E^{2}}{c^{2}}e^{\left[-\frac{2ct}{\text{\text{$\hbar$}}}\left(\boldsymbol{p}+\boldsymbol{p}^{\prime}\right)\right]}+\left\\{\boldsymbol{p}e^{\left[-\frac{ct}{\text{\text{$\hbar$}}}\boldsymbol{p}^{\prime}\right]}+\boldsymbol{p}^{\prime}e^{\left[-\frac{ct}{\text{\text{$\hbar$}}}\boldsymbol{p}\right]}\right\\}^{2}+\frac{E^{2}}{c^{2}}\left\\{e^{\left[-\frac{ct}{\text{\text{$\hbar$}}}\boldsymbol{p}^{\prime}\right]}+e^{\left[-\frac{ct}{\text{\text{$\hbar$}}}\boldsymbol{p}\right]}\right\\}^{2}}\,.$
(38)

Here, $|\mathcal{R}|^{2}$ is the square of the amplitude of quaternionic
reflected wave. Now, the quaternionic reflection coefficient (qRC) can be
defined as

$\displaystyle\mathbb{R}(e_{0},e_{j})=-$
$\displaystyle\frac{J_{\mathcal{R}}}{J_{\mathcal{I}}}\,\,,$ (39)

where $J_{\mathcal{R}}$and $J_{\mathcal{I}}$ are respectively the reflected
and the incident probability current densities that can be determined by
Eq.(26), such as

$\displaystyle J_{\mathcal{R}}\,=$
$\displaystyle\,\,\frac{\boldsymbol{p}}{m}|\mathcal{R}|^{2}+\frac{i\text{$\hbar$}}{2m}\sum_{j=1}^{3}e_{j}\left\llbracket\left(\frac{i\boldsymbol{p}}{\text{\text{$\hbar$}}}-\frac{E}{\text{\text{$\hbar$}}c}\right)\exp\left\\{-\frac{i}{\text{\text{$\hbar$}}}\left[\left(Et+\boldsymbol{p}\cdot\boldsymbol{x}\right)-\left(ict\boldsymbol{p}+\frac{iE}{\text{\text{$c$}}}\boldsymbol{x}+\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right)\right]\right\\}\right.$
$\displaystyle\left.-\left(\frac{i\boldsymbol{p}}{\text{\text{$\hbar$}}}+\frac{E}{\text{\text{$\hbar$}}c}\right)\exp\left\\{\frac{i}{\text{\text{$\hbar$}}}\left[\left(Et+\boldsymbol{p}\cdot\boldsymbol{x}\right)-\left(-ict\boldsymbol{p}-\frac{iE}{\text{\text{$c$}}}\boldsymbol{x}+\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right)\right]\right\\}\right\rrbracket|\mathcal{R}|^{2}\,\,,$
(40)

and

$\displaystyle J_{\mathcal{I}}\,=$
$\displaystyle\,-\frac{\boldsymbol{p}}{m}+\frac{i\text{\text{$\hbar$}}}{2m}\sum_{j=1}^{3}e_{j}\left\llbracket\left(\frac{i\boldsymbol{p}}{\text{\text{$\hbar$}}}+\frac{E}{\text{\text{$\hbar$}}c}\right)\exp\left\\{\frac{i}{\text{\text{$\hbar$}}}\left[\left(Et-\boldsymbol{p}\cdot\boldsymbol{x}\right)-\left(-ict\boldsymbol{p}+\frac{iE}{\text{$c$}}\boldsymbol{x}-\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right)\right]\right\\}\right.$
$\displaystyle\left.-\left(\frac{i\boldsymbol{p}}{\text{\text{$\hbar$}}}-\frac{E}{\text{\text{$\hbar$}}c}\right)\exp\left\\{-\frac{i}{\text{\text{$\hbar$}}}\left[\left(Et-\boldsymbol{p}\cdot\boldsymbol{x}\right)-\left(ict\boldsymbol{p}-\frac{iE}{\text{$c$}}\boldsymbol{x}-\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right)\right]\right\\}\right\rrbracket\,\,,$
(41)

Thus, from Eq.(39) the quaternionic reflection coefficient becomes

$\displaystyle\mathbb{R}(e_{0},e_{j})\,=$
$\displaystyle-\frac{\frac{\boldsymbol{p}}{m}+\frac{i\text{\text{$\hbar$}}}{2m}\sum_{j=1}^{3}e_{j}\left[\left(\frac{i\boldsymbol{p}}{\text{\text{$\hbar$}}}-\frac{E}{\text{\text{$\hbar$}}c}\right)e^{-\frac{i}{\text{\text{$\hbar$}}}\left[Et+\boldsymbol{p}\cdot\boldsymbol{x}-ict\boldsymbol{p}-\frac{iE}{\text{\text{$c$}}}\boldsymbol{x}-\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right]}-\left(\frac{i\boldsymbol{p}}{\text{\text{$\hbar$}}}+\frac{E}{\text{\text{$\hbar$}}c}\right)e^{\frac{i}{\text{\text{$\hbar$}}}\left[Et+\boldsymbol{p}\cdot\boldsymbol{x}+ict\boldsymbol{p}+\frac{iE}{\text{\text{$c$}}}\boldsymbol{x}-\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right]}\right]}{-\frac{\boldsymbol{p}}{m}+\frac{i\text{\text{$\hbar$}}}{2m}\sum_{j=1}^{3}e_{j}\left[\left(\frac{i\boldsymbol{p}}{\text{\text{$\hbar$}}}+\frac{E}{\text{\text{$\hbar$}}c}\right)e^{\frac{i}{\text{\text{$\hbar$}}}\left[Et-\boldsymbol{p}\cdot\boldsymbol{x}+ict\boldsymbol{p}-\frac{iE}{\text{$c$}}\boldsymbol{x}+\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right]}-\left(\frac{i\boldsymbol{p}}{\text{\text{$\hbar$}}}-\frac{E}{\text{\text{$\hbar$}}c}\right)e^{-\frac{i}{\text{\text{$\hbar$}}}\left[Et-\boldsymbol{p}\cdot\boldsymbol{x}-ict\boldsymbol{p}+\frac{iE}{\text{$c$}}\boldsymbol{x}+\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right]}\right]}|\mathcal{R}|^{2}\,,$
(42)

We can split the quaternionic reflection coefficient into two parts, where the
pure scalar part yields

$\displaystyle\mathbb{R}(e_{0},0)$ $\displaystyle\,\,=\,|\mathcal{R}|^{2}\,,$
(43)

while the pure quaternionic part becomes

$\displaystyle\mathbb{R}(0,e_{j})\,$
$\displaystyle=\,\,\sum_{j=1}^{3}e_{j}\left(\frac{\frac{E}{c}\cos\alpha-\boldsymbol{p}\sin\alpha}{\frac{E}{c}\cos\beta-\boldsymbol{p}\sin\beta}\right)\exp\left[-\frac{2E}{\text{$\hbar$}c}\boldsymbol{x}_{j}\right]|\mathcal{R}|^{2}\,,$
(44)

where the parameters
$\alpha=\frac{1}{\text{$\hbar$}}\left[Et+\boldsymbol{p}\cdot\boldsymbol{x}-\left(\boldsymbol{p}\times\boldsymbol{x}\right)\right]$
and
$\beta=\frac{1}{\text{$\hbar$}}\left[Et-\boldsymbol{p}\cdot\boldsymbol{x}+\left(\boldsymbol{p}\times\boldsymbol{x}\right)\right]$.
Therefore, it is conclude that the real quaternionic reflection coefficient
has conventional meaning, however, the pure quaternionic reflection
coefficient shows the additional part associated with quaternionic phase part
and the decaying factor
$\Omega(\boldsymbol{x}_{j})\,\sim\,\exp\left[-\frac{2E}{\text{$\hbar$}c}\boldsymbol{x}_{j}\right]$
along with $e_{j}-$ frames arise from the quaternionic wave function.
Besides, the quaternionic transmission coefficient (qTC) can be identified as

$\displaystyle\mathbb{T}(e_{0},e_{j})\,=$
$\displaystyle\,\,\frac{J_{\mathcal{T}}}{J_{\mathcal{I}}}\,\,,$ (45)

where

$\displaystyle J_{\mathcal{T}}\,=$
$\displaystyle\,-\frac{\boldsymbol{p}^{\prime}}{m}|\mathcal{T}|^{2}+\frac{i\text{\text{$\hbar$}}}{2m}\sum_{j=1}^{3}e_{j}\left\llbracket\left(\frac{i\boldsymbol{p}^{\prime}}{\text{\text{$\hbar$}}}+\frac{E}{\text{\text{$\hbar$}}c}\right)\exp\frac{i}{\text{\text{$\hbar$}}}\left[\left(Et-\boldsymbol{p}^{\prime}\cdot\boldsymbol{x}\right)-\left(-ict\boldsymbol{p}^{\prime}+\frac{iE}{\text{$c$}}\boldsymbol{x}-\left(\boldsymbol{p}^{\prime}\times\boldsymbol{x}\right)_{j}\right)\right]\right.$
$\displaystyle\left.-\left(\frac{i}{\text{\text{$\hbar$}}}\boldsymbol{p}^{\prime}-\frac{E}{\text{\text{$\hbar$}}c}\right)\exp\left\\{-\frac{i}{\text{\text{$\hbar$}}}\left[\left(Et-\boldsymbol{p}^{\prime}\cdot\boldsymbol{x}\right)-\left(ict\boldsymbol{p}^{\prime}-\frac{iE}{\text{$c$}}\boldsymbol{x}-\left(\boldsymbol{p}^{\prime}\times\boldsymbol{x}\right)_{j}\right)\right]\right\\}\right\rrbracket\,|\mathcal{T}|^{2}\,\,.$
(46)

Therefore, by substituting the value of $J_{\mathcal{T}}$ and
$J_{\mathcal{I}},$ the quaternionic transmission coefficient becomes

$\displaystyle\mathbb{T}(e_{0},e_{j})\,=$
$\displaystyle\frac{-\frac{\boldsymbol{p}^{\prime}}{m}+\frac{i\text{\text{$\hbar$}}}{2m}\sum_{j=1}^{3}e_{j}\left[\left(\frac{i\boldsymbol{p}^{\prime}}{\text{\text{$\hbar$}}}+\frac{E}{\text{\text{$\hbar$}}c}\right)e^{\frac{i}{\text{\text{$\hbar$}}}\left[Et-\boldsymbol{p}^{\prime}\cdot\boldsymbol{x}+ict\boldsymbol{p}^{\prime}-\frac{iE}{\text{$c$}}\boldsymbol{x}+\left(\boldsymbol{p}^{\prime}\times\boldsymbol{x}\right)_{j}\right]}-\left(\frac{i\boldsymbol{p}^{\prime}}{\text{\text{$\hbar$}}}-\frac{E}{\text{\text{$\hbar$}}c}\right)e^{-\frac{i}{\text{\text{$\hbar$}}}\left[Et-\boldsymbol{p}^{\prime}\cdot\boldsymbol{x}-ict\boldsymbol{p}^{\prime}+\frac{iE}{\text{$c$}}\boldsymbol{x}+\left(\boldsymbol{p}^{\prime}\times\boldsymbol{x}\right)_{j}\right]}\right]}{-\frac{\boldsymbol{p}}{m}+\frac{i\text{\text{$\hbar$}}}{2m}\sum_{j=1}^{3}e_{j}\left[\left(\frac{i\boldsymbol{p}}{\text{\text{$\hbar$}}}+\frac{E}{\text{\text{$\hbar$}}c}\right)e^{\frac{i}{\text{\text{$\hbar$}}}\left[Et-\boldsymbol{p}\cdot\boldsymbol{x}+ict\boldsymbol{p}-\frac{iE}{\text{$c$}}\boldsymbol{x}+\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right]}-\left(\frac{i\boldsymbol{p}}{\text{\text{$\hbar$}}}-\frac{E}{\text{\text{$\hbar$}}c}\right)e^{-\frac{i}{\text{\text{$\hbar$}}}\left[Et-\boldsymbol{p}\cdot\boldsymbol{x}-ict\boldsymbol{p}+\frac{iE}{\text{$c$}}\boldsymbol{x}+\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right]}\right]}\,|\mathcal{T}|^{2}.$
(47)

The real part of quaternionic transmission coefficient will be contributed as

$\displaystyle\mathbb{T}(e_{0},0)\,=$
$\displaystyle\,\,|\mathcal{T}|^{2}\,\,,$ (48)

while the pure quaternionic part contributes

$\displaystyle\mathbb{T}(0,e_{j})\,=$
$\displaystyle\,\,\sum_{j=1}^{3}e_{j}\left(\frac{\frac{E}{c}\cos\alpha^{\prime}-\boldsymbol{p}^{\prime}\sin\alpha^{\prime}}{\frac{E}{c}\cos\beta^{\prime}-\boldsymbol{p}\sin\beta^{\prime}}\right)\exp\left[-ct\left(\boldsymbol{p}^{\prime}-\boldsymbol{p}\right)\right]\,|\mathcal{T}|^{2}\,\,,$
(49)

where
$\alpha^{\prime}\,=\,\frac{1}{\hbar}\left[Et-\boldsymbol{p^{\prime}}\cdot\boldsymbol{x}+\left(\boldsymbol{p}^{\prime}\times\boldsymbol{x}\right)_{j}\right]$
and
$\beta^{\prime}\,=\,\frac{1}{\hbar}\left[Et-\boldsymbol{p}\cdot\boldsymbol{x}+\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right]$.
Here, the real quaternionic transmission coefficient has usual form, but the
pure quaternionic transmission coefficient has an additional part associated
with the decaying factor
$\exp\left[-ct\left(\boldsymbol{p}^{\prime}-\boldsymbol{p}\right)\right]$ for
$e_{j}-$ axes arise from the quaternionic wave function. Further, if particles
are not to be created or destroyed at the boundary as in a real quaternions,
it strongly follows $\mathbb{R}+\mathbb{T}=1$, but for pure quaternionic case,
the relation violates due to relativistic destroy in quaternionic space. Now,
to check the significance of the quaternionic formalism, in following
subsections we will discuss some important cases for the incident kinetic
energy of particle that compare to the quaternionic step potential.

###  6.1 $\boldsymbol{E>\mathbb{V}_{0}+m_{0}c^{2}}$ : The oscillatory-like
zone

In this case the kinetic energy of particle is greater than the sum of the
step-potential and rest mass energy, then the momentum of the particle for
region $\mathbb{X}\geq 0$ will be positive, i.e.,
$\left.\boldsymbol{P}^{\prime}\right|_{\mathbb{X}\geq
0}=\boldsymbol{p}^{\prime}$. Here, we assume that the lowest feasible value of
step potential becomes the rest mass energy of particle. In order to check the
quaternionic significance with time varying expression, we restrict our
solutions only for time slaps $t\rightarrow 0$ and $t\rightarrow\infty$.

  * •

Case 1: $t\rightarrow 0$. For this case, the square of reflection and
transmission amplitude become:

$\displaystyle|\mathcal{R}|^{2}\,=$
$\displaystyle\,\,\frac{\left(p-p^{\prime}\right)^{2}}{\left(p+p^{\prime}\right)^{2}-4p_{0}^{2}}\,\,\,;\,\,\,\,\,\,|\mathcal{T}|^{2}\,=\,\,\frac{4pp^{\prime}-4p_{0}^{2}}{\left(p+p^{\prime}\right)^{2}-4p_{0}^{2}}\,.$
(50)

Thus, the pure quaternionic form of reflection and transmission coefficients
can be expressed by

$\displaystyle\mathbb{R}(e_{0},0)\,=$
$\displaystyle\,\,|\mathcal{R}|^{2}\,,\,\,\,\,\mathbb{R}(0,e_{j})\,=\,\left(\frac{\frac{E}{c}\cos\alpha_{0}-\boldsymbol{p}\sin\alpha_{0}}{\frac{E}{c}\cos\beta_{0}-\boldsymbol{p}\sin\beta_{0}}\right)\Omega(\boldsymbol{x}_{j})|\mathcal{R}|^{2}\,;$
(51) $\displaystyle\mathbb{T}(e_{0},0)\,=$
$\displaystyle\,\,|\mathcal{T}|^{2}\,,\,\,\,\,\mathbb{T}(0,e_{j})\,=\,\left(\frac{\frac{E}{c}\cos\alpha_{0}^{\prime}-\boldsymbol{p}^{\prime}\sin\alpha_{0}^{\prime}}{\frac{E}{c}\cos\beta_{0}-\boldsymbol{p}\sin\beta_{0}}\right)|\mathcal{T}|^{2}\,,$
(52)

where
$\alpha_{0}=\frac{1}{\text{$\hbar$}}\left[\boldsymbol{p}\cdot\boldsymbol{x}-\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right]$,
$\beta_{0}=\frac{1}{\hbar}\left[-\boldsymbol{p}\cdot\boldsymbol{x}+\left(\boldsymbol{p}\times\boldsymbol{x}\right)_{j}\right]$and
$\alpha_{0}^{\prime}=\frac{1}{\hbar}\left[-\boldsymbol{p^{\prime}}\cdot\boldsymbol{x}+\left(\boldsymbol{p}^{\prime}\times\boldsymbol{x}\right)_{j}\right]$.
The reflection and transmission probability of a particle at the boundary of a
quaternionic step potential is prevalent in pure scalar axes; however, it has
some significant consequences in pure quaternionic axes. Despite the fact that
the kinetic energy of a relativistic particle is greater than the quaternionic
step potential, the probability of some particles reflecting back has a
declining exponential component, whereas transmission has an oscillatory form
for pure quaternionic space.

  * •

Case 2: $t\rightarrow\infty$. In this case, the step potential width of forth-
space component is taken extremely large. Then, the quaternionic reflection
and transmission coefficients become:

$\displaystyle\mathbb{R}(e_{0},0)\,=$
$\displaystyle\,\,\frac{\left(p-p^{\prime}\right)^{2}}{\left(p+p^{\prime}\right)^{2}}\,\,;\,\,\,\,\,\mathbb{R}(0,e_{j})\,=\,\Omega(\boldsymbol{x}_{j})|\mathcal{R}|^{2}\,\,,$
(53) $\displaystyle\mathbb{T}(e_{0},0)\,=$
$\displaystyle\,\,\mathbb{T}(0,e_{j})\,=\,0\,.$ (54)

It is interesting to note that in this case, the quaternionic usual reflection
coefficient is all that exists, and there is no quaternionic transmission for
the potential steps that result from the fourth space component expanding
towards infinity. The particle may therefore act like a classical particle.

###  6.2
$\boldsymbol{\left(\mathbb{V}_{0}-m_{0}c^{2}\right)<E<\left(\mathbb{V}_{0}+m_{0}c^{2}\right)}$:
The tunnelling-like zone

In this case, the kinetic energy of the particle varies within the
quaternionic effective potential, i.e., from $\mathbb{V}_{0}-m_{0}c^{2}$ to
$\mathbb{V}_{0}+m_{0}c^{2}$ (see Fig. 1). Then, for this energy band, one can
have

$\left.\boldsymbol{P^{\prime}}\right|_{\mathbb{X}\geq
0}\,=\,i\boldsymbol{p^{\prime}}$ (55)

where $\boldsymbol{P}^{\prime}$ indicates the imaginary momentum of the
particle for region $\mathbb{X}\geq 0$, so we can simply replace
$i\boldsymbol{p^{\prime}}$ instead of $\boldsymbol{p}$′.

  * •

Case 1: $t\rightarrow 0$. For this case, the square of reflection amplitude
and transmission amplitude modified as:

$\displaystyle|\mathcal{R}|^{2}\,=$
$\displaystyle\,\,\frac{\left(p-ip^{\prime}\right)^{2}}{\left(p+ip^{\prime}\right)^{2}-4p_{0}^{2}}\,\,;\,\,\,\,\,\,\,\,|\mathcal{T}|^{2}\,=\,\frac{4ipp^{\prime}-4p_{0}^{2}}{\left(p+ip^{\prime}\right)^{2}-4p_{0}^{2}}\,\,,$
(56)

which gives

$\displaystyle\mathbb{R}(e_{0},0)\,=$
$\displaystyle\,\,|\mathcal{R}|^{2}\,;\,\,\,\,\mathbb{R}(0,e_{j})\,=\,\,\left(\frac{\frac{E}{c}\cos\alpha_{0}-\boldsymbol{p}\sin\alpha_{0}}{\frac{E}{c}\cos\beta_{0}-\boldsymbol{p}\sin\beta_{0}}\right)\Omega(\boldsymbol{x}_{j})|\mathcal{R}|^{2}\,,$
(57) $\displaystyle\mathbb{T}(e_{0},0)\,=$
$\displaystyle\,\,|\mathcal{T}|^{2}\,;\,\,\,\,\mathbb{T}(0,e_{j})\,=\,\,\left(\frac{\frac{E}{c}\cos\alpha_{0}^{\prime}-i\boldsymbol{p}^{\prime}\sin\alpha_{0}^{\prime}}{\frac{E}{c}\cos\beta_{0}-\boldsymbol{p}\sin\beta_{0}}\right)|\mathcal{T}|^{2}\,.$
(58)

However, for time $t\rightarrow\tau$ the pure quaternionic transmission
coefficient varies as

$\displaystyle\left.\mathbb{T}(0,e_{j})\right|_{t\rightarrow\tau}\,=\,\,\mathbb{T}(0,e_{j})\,$
$\displaystyle e^{-\xi\,\tau}\,,$ (59)

where $\xi=c\boldsymbol{p}^{\prime}-c\boldsymbol{p}$ indicates the change of
relativistic kinetic energy of a moving particle from region $\mathbb{X}<0$ to
$\mathbb{X}\geq 0$. As a result of Eq.(59), the transmission coefficient for a
pure quaternionic framework decays exponentially with time. Thus, the
probability of identifying particles in $\mathbb{X}>0$ within such a
tunnelling-like zone can be estimated by the fourth-space component of pure
quaternionic space.

  * •

Case 2: $t\rightarrow\infty$. The reflection and transmission coefficients for
the infinitely extended step potential width due to the quaternionic fourth-
space component are

$\displaystyle\mathbb{R}(e_{0},0)\,=$
$\displaystyle\,\,|\mathcal{R}|^{2}\,\,\equiv\,\,\frac{\left(p-ip^{\prime}\right)^{2}}{\left(p+ip^{\prime}\right)^{2}}\,\,\,;\,\,\,\,\,\mathbb{R}(0,e_{j})\,=\,\Omega(\boldsymbol{x}_{j})|\mathcal{R}|^{2}\,\,,$
(60) $\displaystyle\mathbb{T}(e_{0},0)\,=$
$\displaystyle\,\,\mathbb{T}(0,e_{j})\,=\,0\,.$ (61)

Essentially, as shown in subsection 6.2, only the quaternionic usual
reflection coefficient is present in this situation, and no quaternionic
transmission takes place for any hypothetical steps that result in the fourth
space component stretching towards infinity.

###  6.3 $\boldsymbol{E<\left(\mathbb{V}_{0}-m_{0}c^{2}\right)}$: The Klein-
like zone

This zone is crucial because if the particle’s kinetic energy is significantly
lower than the height of the tunnelling-like zone, inappropriate particle
reflection will have appeared. In order to analyze the Klein paradox behavior
of relativistic particles in a quaternionic frame, let us write the momentum
of the particle for region $\mathbb{X}\geq 0$, which becomes negative as

$\left.\boldsymbol{P}^{\prime}\right|_{\mathbb{X}\geq
0}=\,\,-\boldsymbol{p}^{\prime}\,\,.$ (62)

To examine the probability of particles in the regions $\mathbb{X}<0$ and
$\mathbb{X}\geq 0$, we can place this opposite momentum value in Eqs.(39) and
(45) as the following manner:

  * •

Case 1: $t\rightarrow 0$. In this instance, the square of the reflection and
transmission amplitudes of the quaternionic wave is adjusted as follows:

$\displaystyle|\mathcal{R}|^{2}\,\,=\,\,\frac{\left(p+p^{\prime}\right)^{2}}{\left(p-p^{\prime}\right)^{2}-4p_{0}^{2}}\,\,;\,\,\,\,\,\,\,\,\,|\mathcal{T}|^{2}\,\,=$
$\displaystyle\,\,\frac{-4\left(pp^{\prime}+p_{0}^{2}\right)}{\left(p-p^{\prime}\right)^{2}-4p_{0}^{2}}\,\,.$
(63)

Hence, the quaternionic reflection coefficients i.e.,
$\mathbb{R}(e_{0},0),\,$$\mathbb{R}(0,e_{j})$ become:

$\displaystyle\mathbb{R}(e_{0},0)\,=$
$\displaystyle\,\,|\mathcal{R}|^{2}\,\,\,;\,\,\,\,\,\,\mathbb{R}(0,e_{j})\,=\,\,\left(\frac{\frac{E}{c}\cos\alpha_{0}-\boldsymbol{p}\sin\alpha_{0}}{\frac{E}{c}\cos\beta_{0}-\boldsymbol{p}\sin\beta_{0}}\right)\Omega(\boldsymbol{x}_{j})|\mathcal{R}|^{2}\,\,.$
(64)

Since we see that $|\mathcal{R}|^{2}>1$, it is clear that the reflection
coefficient increases and exceeds value one in both pure scalar and pure
quaternionic circumstances that is providing an incompatible outcome for the
quaternionic Klein paradox. A reflection coefficient larger than one caused by
opposite momentum due to the quaternionic quantum fluctuations implies more
particles will reflect that fall on it. However, the transmission coefficient
changes to:

$\displaystyle\mathbb{T}(e_{0},0)\,=$
$\displaystyle\,\,|\mathcal{T}|^{2}\,\,\,;\,\,\,\,\,\,\mathbb{T}(0,e_{j})\,=\,\left[\frac{\frac{E}{c}\cos\alpha_{0}^{\prime}+\boldsymbol{p}^{\prime}\sin\alpha_{0}^{\prime}}{\frac{E}{c}\cos\beta_{0}-\boldsymbol{p}\sin\beta_{0}}\right]|\mathcal{T}|^{2}\,\,.$
(65)

The relevance of the quaternionic Klein paradox is provided by the fact that
$|\mathcal{T}|^{2}<0$, which means that the quaternionic transmission
coefficients are negative for both pure scalar and pure quaternionic
conditions.

  * •

Case 2: $t\rightarrow\infty$. For this situation, the reflection and the
transmission coefficients are expressed by

$\displaystyle\mathbb{R}(e_{0},0)\,=$
$\displaystyle\,\,|\mathcal{R}|^{2}\,\equiv\,\frac{\left(p+p^{\prime}\right)^{2}}{\left(p-p^{\prime}\right)^{2}}\,\,\,;\,\,\,\,\,\,\,\,\mathbb{R}(0,e_{j})\,=\,\Omega(\boldsymbol{x}_{j})|\mathcal{R}|^{2}\,\,,$
(66) $\displaystyle\mathbb{T}(e_{0},0)\,=$
$\displaystyle\,\,\mathbb{T}(0,e_{j})=\,0\,.$ (67)

Therefore, the probability of finding the particles in the reflected case is
still greater than one, as explained in case $t\rightarrow 0$, but it impacts
zero for the quaternionic transmitted probability when we increase the fourth
dimension of space in the potential step up to infinity.

Hence, it is claimed that by reducing the step potential width at a small
scale, the fundamental Klein-Gordon wave equation in quaternionic
($3+1$)$-$fields may be extended to explain the dynamics of relativistic
particles in finite potential barrier.

##  7 Conclusion

The proposed research examines the Klein-Gordon wave equation in the context
of quaternionic step potential by employing the $(3+1)-$dimensional
quaternionic wave function. The quaternionic generalization of the Klein-
Gordon equation has been established for both scalar and vector fields. We
reviewed the quaternionic continuity-like equation that connected probability
density and probability current density for pure scalars, pure vectors, and
their mixed fields. It is intriguing that the combined field results from the
mutual interaction of the real and pure quaternionic fields. Furthermore, the
amplitudes of the incident, reflected, and transmitted waves for the
quaternionic four-potential step have been analyzed. It is emphasized that the
pure scalar component of the quaternionic incident, reflected, and transmitted
waves is analogous to conventional waves; however, the additional or pure
vector component of these waves appears due to the presence of a quaternionic
structure. Correspondingly, we found the reflection and transmission
coefficients associated with the quaternionic four-potential step, which
demonstrate the probability of the existence of waves (or particles) in
different regions of the step potential problem. Surprisingly, the
quaternionic value of the reflected and transmitted coefficients is determined
by the particle’s four-momentum. Furthermore, we have addressed the
quaternionic interpretation for the oscillatory, tunnelling, and Klein paradox
instances in which the incoming wave-function interacts with the potential
step via the quaternionic momentum of the particle. It has also been claimed
that the physical conservation law $\mathbb{R}+\mathbb{T}=1$ does not hold for
pure quaternionic fields owing to non-commutativity, although it holds for
pure scalar fields. Subsequently, it has been decided that, in order to
account for the possibility of the quaternionic Klein paradox, the particle
momentum in the step potential region ($\mathbb{X}\geq 0$) must be opposite to
that region ($\mathbb{X}<0$), which suggests that the kinetic energy of the
incoming wave (or particle) becomes negative. For the quaternionic Klein
paradox, the quaternionic reflection coefficient becomes solely higher than
one and the quaternionic transmission coefficient becomes simply less than
zero. As a result, the current quaternionic ($3+1$) formalism is sufficiently
consistent to explain the relativistic quantum phenomena of the contemporary
era.

## References

  * [1] W. R. Hamilton, “Elementary of Quaternions”, Vol. I & Vol. II, Chelsea Publishing, New York, (1969).
  * [2] P. A. Bolokhov, “Quaternionic wave function”, Int. J. Mod. Phys. A, 34 (2019), 1950001\.
  * [3] D. Finkelstein, J. M. Jauch and D. Speiser, “Quaternionic representations of compact groups”, J. Math. Phys., 4 (1963), 136.
  * [4] S. L. Adler, “Quaternionic Quantum Mechanics and Quantum Fields”, Oxford University Press, (1995).
  * [5] R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics”, Rev. Mod. Phys., 20 (1948), 367.
  * [6] A. Das, S. Okubo and S. A. Pernice, “Higher-dimensional SUSY quantum mechanics”, Mod. Phys. Lett. A, 12 (1997), 581.
  * [7] L. Silberstein, “Quaternionic form of Relativity”, Philos. Mag., 23 (1912), 790.
  * [8] S. Giardino, “Non-anti-Hermitian Quaternionic Quantum Mechanics”, Adv. Appl. Clifford Algebras, 28 (2018), 19.
  * [9] S. Giardino, “Virial theorem and generalized momentum in quaternionic quantum mechanics”, Eur. Phys. J. Plus, 135 (2020), 114.
  * [10] S. Giardino, “Quaternionic quantum mechanics in real hilbert space”, J. Geom. Phys. 158 (2020), 103956.
  * [11] S. Giardino, “Quaternionic Klein–Gordon equation”, Eur. Phys. J. Plus, 136 (2021), 612.
  * [12] S. Rawat, “Quaternionic Klein-Gordon equation in the electromagnetic field”, Eur. Phys. J. Plus, 137 (2022), 620.
  * [13] A. J. Davies, “Quaternionic Dirac equation”, Phys. Rev., 41 (1990), 2628.
  * [14] S. D. Leo and P. Rotelli, “The Quaternionic Dirac Lagrangian”, Mod. Phys. Lett. A, 11 (1996), 357.
  * [15] M. Carmeli, “Field theory on R×S3 topology 1: The Klein-Gordon and Schrodinger equations”, Found. Phys., 15 (1983), 175
  * [16] L. Silberstein, “Quaternionic form of Relativity”, Philos. Mag., 23 (1912), 790.
  * [17] S. De Leo, “Quaternions and special relativity”, J. Math. Phys., 37 (1996), 2955\.
  * [18] W. Sprossig, “Quaternionic operator methods in fluid dynamics”, Adv. Appl. Clifford Algebras, 18 (2008), 963.
  * [19] L. P. Horwitz and L. C. Biedenharn, “Quaternion quantum mechanics: second quantization and gauge fields”, Ann. Phys., 157 (1984) 432
  * [20] S. De Leo, G. C. Ducati and C. C. Nishi, “Quateraionic potentials in non-relativistic quantum mechanics”, J. Phys. A: Math Gen., 35 (2002), 5411.
  * [21] A. J. Davies and B. H. J. McKellar, “Nonrelativistic quaternionic quantum mechanics in one dimension”, Phys. Rev. A, 40 (1989), 4209.
  * [22] S. Rawat and O. P. S. Negi, “Quaternion dirac equation and supersymmetry”. Int. J. Theor. Phys., 48 (2009), 2222.
  * [23] S. Ulrych, “Higher spin quternion waves in the Klein Gordon Theory”, Int. J. Theor. Phys. 52 (2013), 279.
  * [24] B. C. Chanyal, “A relativistic quantum theory of dyons wave propagation”, Can. J. Phys., 95 (2017), 1200.
  * [25] B. C. Chanyal, “A new development in quantum field equations of dyons”, Can. J. Phys., 96 (2018), 1192.
  * [26] B. C. Chanyal, “Quaternionic approach on the Dirac–Maxwell, Bernoulli and Navier–Stokes equations for dyonic fluid plasma”, Int. J. Modern Phys. A, 34 (2019), 1950202.
  * [27] B. C. Chanyal, M. Pathak, “Quaternionic approach to dual Magneto-hydrodynamics of dyonic cold plasma”, Adv. High Energy Phys. 2018 (2018) Article ID 7843730, 13.
  * [28] B. C. Chanyal, “On the quaternion transformation and field equations in curved space-time”, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 93 (2023), 185.
  * [29] S. De Leo and S. Giardino, “Dirac solutions for quaternionic potentials”, J. Math. Phys., 55 (2014), 022301.
  * [30] S. De Leo, G. Ducati and S. Giardino, “Quaternionic dirac scattering”, J. Phys. Math., 6 (2015),1 000130.
  * [31] S. De Leo and G. Ducati, “Quaternionic differential operators”, J. Math. Phys., 42 (2001), 2236\.
  * [32] O. Klein, “The reflection of electron by a potential jump according to Dirac’s relativistic dynamics”, Zeit. Phys., 53 (1929), 157.
  * [33] A. Calogeracos and N. Dombey, “History and physics of the Klein paradox”, Contemp. Phys., 40 (1999), 313.
  * [34] H. Y. Wang, “Solving Klein’s paradox”, J. Phys. Commun., 4 (2020), 1.
  * [35] A. Hansen and F. Ravndal, “Klein’s paradox and its resolution”, Phys. Scr., 23 (1981), 1036\.
  * [36] B. C. Chanyal and S. Karnatak, A comparative study of quaternionic rotational Dirac equation and its interpretation, Int. J. Geom. Methods Mod. Phys., 17 (2020) 2050018.
  * [37] S. Bhatt and B. C. Chanyal, Generalized quaternionic free rotational Dirac equation and spinor solutions in the electromagnetic field, Int. J. Geom. Methods Mod. Phys., 19 (2022) 2250103.