# Thermodynamic Limits of Photon-Multiplier Luminescent Solar Concentrators

Tomi K. Baikie  Arjun Ashoka  Akshay Rao  Neil C. Greenham<EMAIL_ADDRESS>Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, UK

###### Abstract

Luminescent solar concentrators (LSCs) are theoretically able to concentrate
both direct and diffuse solar radiation with extremely high efficiencies.
Photon-multiplier luminescent solar concentrators (PM-LSCs) contain
chromophores which exceed 100% photoluminescence quantum efficiency. PM-LSCs
have recently been experimentally demonstrated and hold promise to outcompete
traditional LSCs. However, we find that the thermodynamic limits of PM-LSCs
are different and are sometimes more extreme relative to traditional LSCs. As
might be expected, to achieve very high concentration factors a PM-LSC design
must also include a free energy change, analogous to the Stokes shift in
traditional LSCs. Notably, unlike LSCs, the maximum concentration ratio of a
PM-LSC is dependent on brightness of the incident photon field. For some
brightnesses, but equivalent energy loss, the PM-LSC has a greater maximum
concentration factor than that of the traditional LSC. We find that the
thermodynamic requirements to achieve highly concentrating PM-LSCs differ from
traditional LSCs. The new model gives insight into the limits of concentration
of PM-LSCs and may be used to extract design rules for further PM-LSC design.

††preprint: APS/123-QED

##  I Introduction

There is currently much interest in improving the yield of solar energy
capture technologies, by making better use of the incoming solar radiation.
One way to improve the solar cell effectiveness is to concentrate incoming
solar irradiation [1, 2, 3]. But in many regions of the world, the majority of
the incident light is diffuse, but diffuse light cannot be concentrated by
traditional image-preserving optical techniques, due to the limits established
by the second law of thermodynamics [4, 5]. Traditional luminescent solar
concentrators (LSC) offer an elegant way around this problem. An LSC is
typically constructed from plastic or glass sheets with embedded chromophores,
organic dyes or semiconductor nanocrystals, which efficiently absorb and emit
light (Fig 1). There is typically a free energy change which offsets the
entropic loss when concentrating diffuse light. The LSC can thus concentrate
both direct and diffuse light.

In traditional LSCs, a high-energy absorbed photon is converted to a lower-
energy photon by the absorption and emission of a chromophore, inducing
thermal loss in the form of a Stokes shift. The thermodynamics and efficiency
limits of these systems have been previously described in detail [1, 6, 7].
Recently, there have been proposals and experimental demonstrations of a new
class of photon-multiplier luminescent solar concentrators (PM-LSCs), where
high-energy photons are converted to two low-energy photons via processes such
as quantum cutting in lanthanide-doped nanocrystals. Chromophores exhibiting
up to 180% PLQE have been experimentally demonstrated [8, 9]. Such systems
could enable photoluminescence quantum efficiencies (PLQE) as high as 200%.
The low reabsorption and high PLQE could make such systems ideal candidates
for LSCs. There is currently no theoretical treatment of the thermodynamic
limits of such a system, which are often useful to benchmark real device
efficiencies against [10].

Here, we present a thermodynamic model which gives insight into the
fundamental limits of PM-LSCs. The maximum concentration performance of PM-
LSCs is linked to brightness of the incoming photon field, which is not the
case for traditional LSCs. Further, the maximum concentration factors
associated with PM-LSCs are different, and at high brightness are more
extreme, relative to their traditional counterparts.

Figure 1:  Schematic of operation of a photon-multiplier luminescent solar
concentrators (PM-LSCs). As an input beam, $B_{1}$, irradiates the device, a
chromophore absorbs and emits light at a longer wavelength. In a PM-LSC,
maximally two photons are emitted for every incoming photon. Supposing
isotropic emission, the number of photons lost through the escape cone is
dependent on the ratio of the refractive indices of the outside medium
($n_{1}$) to the medium of the LSC ($n_{2}$). The photons which are emitted
within the LSC are described by the second concentrated beam, $B_{2}$, which
in proper operation impinges on a photovoltaic mounted on the edge of the LSC.

##  II Etendue and Entropy

A beam of radiation, passing through an absorbing and luminescent medium, can
be characterized by a temperature, chemical potential and etendue [11, 12].
For a beam with ray direction within solid angle $\textrm{d}\Omega$ passing
through an area $\textrm{d}A$, the element of etendue $\textrm{d}\xi$ is given
by

$\textrm{d}\xi=n^{2}\cos(\theta)\textrm{d}\Omega\textrm{d}A,$ (1)

where $\theta$ is the angle between the direction of propagation and the
normal to $\textrm{d}A$, and $n$ is the refractive index of the medium. The
etendue of a beam propagating through a clear and transparent medium is
conserved. This conservation of etendue can be interpreted as the Liouville’s
theorem of classical mechanics, applied to photon propagation along rays in
geometrical optics. Should absorption and remission occur, the etendue change
is then a measure of the free energy loss. At moderate intensities of light,
the thermodynamics of the system can be described by a two-dimensional gas
where the number of dimensions arises from the two angular coordinates
required to specify the direction of the beam [13].

Notably, if entropy and etendue are constant, any concentration system which
decreases the illuminated area $A$ must increase the angular divergence of the
beam. In the case of diffuse light, the angular divergence is already maximal,
and thus diffuse radiation cannot be concentrated with a system using only
geometrical optics. In a traditional LSC, the change in enthalpy from heat
dissipation associated with the Stokes shift can offset the entropic loss,
allowing for the concentration of diffuse light with an associated energetic
penalty.

The Heisenberg uncertainty principle, considered in the absence of
diffraction, places a limit on the spread of the beam wavevectors in momentum
space ; $\textrm{d}x\textrm{d}p_{x}\geq h$[12]. This condition naturally
requires that only a single quantum state occupies the phase-space volume
$\textrm{d}x\ \textrm{d}y\ \textrm{d}z\ \textrm{d}p_{x}\ \textrm{d}p_{y}\
\textrm{d}p_{z}$. A suitable choice of coordinate system gives the beam area,
$\textrm{d}A$, as $\textrm{d}x\ \textrm{d}y$. The etendue element can then be
written as

$\textrm{d}\xi=\frac{n^{2}}{k^{2}}\ \textrm{d}x\ \textrm{d}y\ \textrm{d}k_{x}\
\textrm{d}k_{y}.$ (2)

The volume element $\textrm{d}x\ \textrm{d}y\ \textrm{d}k_{x}\
\textrm{d}k_{y}$ in the phase space of variables $x,y,k_{x},k_{y}$ is equal to
$k^{2}\textrm{d}\xi/n^{2}$, where $n$ is the refractive index, and contains a
single quantum state which can be occupied by a photon in a beam. For a system
of $N$ photons, and allowing for two directions of polarization, we define

$G=\frac{2k^{2}}{n^{2}}\xi=\frac{2\nu^{2}}{c^{2}}\xi,$ (3)

which can be interpreted as the number of quantum states within a beam with
frequency $\nu=ck/n$ and etendue $\xi$. These $N$ indistinguishable photons
are distributed over the $G$ states according to,

$W=\frac{(G+N)!}{G!N!}\approx\frac{(G+N)^{G+N}}{G^{G}N^{N}},$ (4)

where the Stirling’s approximation has been used to obtain the second result.
Invoking the ergodic hypothesis that each microstate is equally probable, the
entropy of the beam, $S$, is the natural logarithm of the number of
microstates, $W$, multiplied by the Boltzmann constant $k_{B}$,

$\begin{split}S&=k_{B}\ln(W)\\\
&=k_{B}\frac{2\nu^{2}}{c^{2}}\xi\Bigg{(}\bigg{(}1+\frac{N}{G}\bigg{)}\ln\bigg{(}1+\frac{N}{G}\bigg{)}-\frac{N}{G}\ln\bigg{(}\frac{N}{G}\bigg{)}\Bigg{)}.\\\
\end{split}$ (5)

Following Yablonovitch’s treatment, the entropy per photon, $s$, is the number
differential of the entropy of the beam [1],

$\begin{split}s=\frac{\textrm{d}S}{\textrm{d}N}&=k_{B}\ln\bigg{(}1+\frac{2\nu^{2}}{c^{2}}\frac{\xi}{N}\bigg{)}.\end{split}$
(6)

In the limit of maximum efficiency, where the loss of one photon from the
incoming beam immediately results in the addition of two photons to the
concentrated beam, the change in entropy of the concentrated photon beam then
becomes

$\Delta S=2k_{B}\ln\bigg{(}1+\frac{2\nu^{2}}{c^{2}}\frac{\xi}{N}\bigg{)}.$ (7)

Using the thermodynamic definition of entropy, $\delta Q/T$, rather than the
statistical argument developed here, Equation 7 can be equivalently recovered
by assuming that the photon gas is an effective heat bath and $\delta Q$ is
the addition of 2 photons of energy $h\nu$.

##  III Brightness

Radiation will come into thermal equilibrium with the chromophore provided
that there is fast thermal equilibration among excited states [1, 12]. As
there are no first-order photon-photon interactions, photon beams cannot be in
a true thermal equilibrium. Nevertheless, photon beams are well defined by
thermodynamic parameters such as temperature, $T$, and chemical potential,
$\mu$ [11]. The radiation emitted by a black body has zero chemical potential,
however, quasi-blackbody radiation, where radiation is emitted over a
restricted range, may have a non-zero chemical potential. Planck derives the
intensity of a monochromatic beam as

$I=\frac{h\nu^{3}n^{2}}{c^{2}}\frac{1}{e^{\frac{\mu-h\nu}{k_{B}T}}-1},$ (8)

with has units of energy per unit area, where $k_{B}$ is the Boltzmann
constant and $T$ is the temperature of the medium in which emission from the
chromophore takes place [14, Eq. 300]. As radiation in thermodynamic
equilibrium is isotropic and unpolarised we can consider average chromophore
absorption over $4\pi$ solid angle. Dividing Equation 8 by the energy of the
photon ($h\nu$), we determine the photon flux of a beam as

$B=\frac{8\pi n^{2}\nu^{2}}{c^{2}}\frac{1}{e^{\frac{\mu-h\nu}{k_{B}T}}-1}.$
(9)

Equation 9 is often referred to as the brightness of the beam and has units of
photons per second, per unit bandwidth, per 4$\pi$ solid angle, per unit area
[14, 1, 11, 15]. For monochromatic brightness we can write
$\frac{\xi}{N}=\frac{4\pi n^{2}}{B}$. Where the input beam has some frequency
range, the treatment presented by Markvart may be used [12]. Brightness, like
etendue, is conserved along the path of the beam in a perfectly transparent
material. The transformation of one quasi-equilibrium beam into another
introduces irreversibility, bringing about entropy generation.

##  IV Concentration Limit

The gain in entropy of the concentrated beam must be acquired from the free
energy change associated with the photon-multiplicative process. For a
traditional LSC with a chromophore of unity photoluminescence quantum
efficiency, the concentration factor, C, is defined as the ratio of the
brightnesses of the outgoing field, $B_{2}$, and the incoming field, $B_{1}$,

$C\equiv\frac{B_{2}}{B_{1}}\leq\frac{\nu_{2}^{2}}{\nu_{1}^{2}}\exp\bigg{(}\frac{h\big{(}\nu_{1}-\nu_{2}\big{)}}{k_{B}T}\bigg{)},$
(10)

where $\nu_{1}$ and $\nu_{2}$ are the frequencies of the input photon and
output photon, respectively [1]. The concentration limit of the PM-LSC can be
determined from the second law of thermodynamics. Including the entropy source
from the thermal loss, $h(\nu_{1}-2\nu_{2})$, the second law can be written

$\underbrace{2k_{B}\ln\Bigg{(}1+\alpha\frac{\nu_{2}^{2}}{B_{2}}\Bigg{)}}_{\Delta
S_{2}}+\frac{h(\nu_{1}-2\nu_{2})}{T}-\underbrace{k_{B}\ln\Bigg{(}1+\alpha\frac{\nu_{1}^{2}}{B_{1}}\Bigg{)}}_{\Delta
S_{1}}\geq 0,$ (11)

where $\alpha=8\pi n^{2}/c^{2}$ and $\Delta S_{1}$ and $\Delta S_{2}$
correspond to the entropy changes in the incident photon field and the
generated photon field, respectively. Assuming the LSC is operating in a
normal terrestrial environment, where $\alpha\nu^{2}/B\gg 1$, the ones in the
logarithms can be ignored, and a concentration factor for PM-LSCs can be
determined by rearranging Equation 11;

$\begin{split}\frac{B_{2}^{2}}{B_{1}}&\leq\frac{8n^{2}\pi}{c^{2}}\frac{\nu_{2}^{4}}{\nu_{1}^{2}}\exp\bigg{(}\frac{h(\nu_{1}-2\nu_{2})}{k_{B}T}\bigg{)}.\\\
\end{split}$ (12)

Hence the concentration factor is given by

$\begin{split}\frac{B_{2}}{B_{1}}&\leq\sqrt{\frac{8n^{2}\pi}{B_{1}c^{2}}\frac{\nu_{2}^{4}}{\nu_{1}^{2}}\exp\bigg{(}\frac{h(\nu_{1}-2\nu_{2})}{k_{B}T}\bigg{)}}.\\\
\end{split}$ (13)

Figure 2:  Maximum concentration factor as function of chromophore absorption
operating under terrestrial conditions. The dotted and solid lines gives the
maximum concentration factor, $C$, of the traditional LSC and PM-LSC,
respectively, at room temperature ($T=300$ K) with $n=1$, when illuminated
with light of brightness $10^{9}$ photons per second per unit bandwidth per
unit area per $4\pi$ solid angle. This brightness roughly approximates a broad
chromophore absorption over the terrestrial solar spectrum (see Appendix A).
Emission for both the PM-LSC and the traditional LSC are fixed by
$h\nu_{2}=1.12$ eV, to match the bandgap of silicon photovoltaics. In the PM-
LSC, the onset of the photon-multiplication process occurs at
$\nu_{1}=2\nu_{2}$. In normal terrestrial conditions, the PM-LSC may
concentrate at $\nu_{1}=2\nu_{2}$ since the entropic change associated with
the removal and addition of photons at different wavelengths ($\Delta
S_{2}-\Delta S_{1}$) is greater than zero. When absorbing higher energy
photons, $\nu_{1}>2\nu_{2}$, the maximum concentration factor increases as the
thermal loss is increases, analogous to the Stokes shift in the traditional
LSC.

Surprisingly, we find that for PM-LSCs the concentration limit is dependent on
input brightness. Comparing the concentration limit of a PM-LSC, Equation 13,
to that of a traditional LSC, Equation 10, we find similar design
requirements, but also some important distinctions [1].

In a traditional LSC, when the Stokes shift is zero ($\nu_{1}=\nu_{2}$), there
will be no associated thermal loss as $h(\nu_{1}-\nu_{2})/T=0$, and the
difference in the entropic terms associated with the addition and removal of
the photon will equal, $\Delta S_{2}-\Delta S_{1}=0$, thus the LSC is unable
to concentrate. However, in the PM-LSC, particularly at low brightnesses, may
concentrate, even if there is no thermal loss, i.e. $h(\nu_{1}-2\nu_{2})/T=0$.
Even at the energy conservation limit, $\nu_{1}=2\nu_{2}$, there remains a
source of entropy from the addition and removal of photons of different
wavelengths, $\Delta S_{2}-\Delta S_{1}$) which is sufficient to allow for
concentration.

To increase the concentration limit of a PM-LSC, or to enable effective
concentration at higher brightnesses, an additional thermal loss is required
to satisfy the inequality in Equation 13, analogous to the Stokes shift in the
traditional LSC. As depicted in Figure 2, the concentration factor is
exponentially dependent on the thermal loss in PM-LSCs. Figure 2 assumes that
both the PM-LSC and LSC chromophore emission is given by $h\nu_{2}=1.12$ eV,
and thus optimised for the silicon photovoltaic bandgap ($E_{g}=1.12$ eV).
Both the traditional LSC and the PM-LSC are assumed to have the same matrix
medium with a refractive index of $n=1$. In the case of PM-LSCs, only photons
of energy of $h\nu_{1}\geq 2.48$ eV may undergo a PM processes to comply with
the constraints imposed by the conservation of energy. The input brightness,
$B_{1}$, is fixed at $10^{9}$ photons per second per unit bandwidth per unit
area per $4\pi$ solid angle. This brightness represents a broad absorption
over the terrestrial solar spectrum for the PM-LSC, though there exists
difficulties in determining this value when approximating solar radiation over
some narrow bandwidth (see Appendix A). Care must be taken when modelling much
greater brightnesses, for example where the input solar flux has been
concentrated before entry into the LSC, or for extremely small bandgap
materials, where Equation 13 must be reformed, as $\alpha\nu^{2}/B\gg 1$ is no
longer valid (see Appendix B).

We have identified that to achieve very high concentration factors, as for
traditional LSCs, a PM-LSC design may also include a thermal loss.
Nevertheless, for some brightnesses, but equivalent thermal loss, the PM-LSC
has a greater maximum concentration factor than that of the traditional LSC.

Figure 3: Maximum concentration factor at room temperature for a traditional
LSC (dotted) and a PM-LSC (solid) as function of incident brightness. Both the
PM-LSC and the traditional LSC concentration limit have emission at
$h\nu_{2}=1.12$ eV, optimised for silicon photovoltaics. The traditional LSC
has a chromophore with a Stokes shift of $0.24$ eV ($\lambda_{1}=912$ nm)
which gives a brightness independent maximum concentration factor $C=7280$.
The PM-LSC is assumed to be 200 percent efficient. The PM-LSC has a thermal
loss matching that of the traditional LSC, i.e. $h\nu_{1}=2\times
1.12+0.24=2.48$ eV input, corresponding to monochromatic incident light of
wavelength $\lambda_{1}=500$ nm.

Figure 3 outlines the effect of input brightness on concentration factor. Here
again, the emission is optimised for a silicon photovoltaic bandgap
($h\nu_{2}=1.12$ eV). In this case, however, the thermal loss is fixed at
$0.24$ eV for both the traditional LSC and the PM-LSC. The fixed thermal loss
defines the monochromatic photon absorption at $\lambda_{1}=912$ nm in the
traditional LSC and at $\lambda_{1}=500$ nm in the PM-LSC. For this specific
thermal loss, only weak brightnesses allow for a maximum PM-LSC concentration
factor greater than the traditional LSCs. Although the maximum concentration
factor for PM-LSCs may increase more rapidly with increasing thermal loss
relative to the traditional LSC, at very high brightness the PM-LSC cannot
reach the same maximum concentration ratios as that of the traditional LSC.

Figure 4: Maximum concentration as function of PM efficiency and input
brightness. The blue surface represents the maximum concentration factor for a
PM-LSC with a variable PM efficiency, $\gamma$, under different incident
brightness. $\gamma$ varies from 1, which would be the maximum efficiency of a
traditional LSC, to 2, the maximum efficiency of PM-LSCs considered here. The
emission frequency for the PM-LSC is fixed at $h\nu_{2}=1.12$ eV, to match
that of silicon photovoltaics. Incident illumination is described by a
monochromatic beam with wavelength $h\nu_{1}=1.12\gamma+0.15$ eV. The input
brightness is plotted between $10^{8}$ to $10^{12}$ photons per second, per
unit bandwidth, per unit area, per $4\pi$ solid angle.

Significant research effort has been expended to improve the efficiencies and
understand the mechanisms of photon-multiplier technologies [16, 17]. As such,
it is of general interest to determine how material design improvements may
impact on LSC uses and design. Figure 4 plots PM-LSC concentration as a
function of $\gamma$, which is a measure of the PM-system efficiency between
$1\leq\gamma\leq 2$ (see Appendix C for full expression). A maximally
efficient traditional LSC chromophore will have $\gamma=1$, meaning one photon
absorbed will result in one photon emitted by the chromophore, whereas a PM-
LSC with $200\%$ photon conversion efficiency has $\gamma=2$. We note that
while the notion of a fractional $\gamma$ is not physical in the single photon
limit, in the thermodynamic limit which we study, it is purely a ratio of
underlying photon-yielding processes. For example, in organic semiconductors,
the interplay between singlet fission or inter-system crossing pathways for
photoexcited singlets sets the effective $\gamma$ of the system, with pure
singlet fission yielding $\gamma=2$ and pure inter-system crossing yielding
$\gamma=1$, assuming the triplets eventually yield luminescent species of
interest[16].

At low brightness, even a small improvement in $\gamma$, from $1$ to $1.2$
would offer a two-fold increase in the maximum concentration factor of PM-
LSCs, implying that moving away from traditional LSCs to even weakly efficient
PM process yields gains in the concentration factor. For the highest
brightnesses, however, a diminishing return on the investment of improving
$\gamma$ is made. Taking the example of singlet fission photon-multiplier
systems, this suggests that even a small improvement in the singlet fission
process at low brightnesses could lead to impressive improvements in the
maximum concentration factors of PM-LSCs. This also suggests PM-LCSs may play
an important role in low brightness energy harvesting systems.

Further, this work suggests that exothermic singlet fission chromophores may
lead to higher maximum concentration factors relative to endothermic singlet
fission chromophores [16]. Without the inclusion of another heat source,
endothermic singlet fission chromophores would act to reduce the entropic loss
in Equation 11 and hence reduce the concentration limit.

Practical demonstrations of PM-LSCs utilise quantum cutting, where ytterbium
doped perovskite nanocrystals feature a photoluminescence quantum yield
approaching 200% and virtually zero self-absorption loss [8, 9]. These systems
show great promise for PM-LSCs, however, the ytterbium-doped perovskite
chromophores exhibit long-lived excited states on the order of milliseconds.
These systems violate our assumption that the chromophore exhibits quick
electronic recovery to a photon accepting state [18]. While these systems
exhibit long lived excited states, they will not reach the concentration
limits described here.

In practice, both PM-LSCS and traditional LSCs are far from being limited by
the entropic arguments put forward here. Record demonstrations of traditional
LSCs achieve concentration ratios on the order of 1-10 [19]. Although at high
brightness the PM-LSC concentration limit is significantly below that of
traditional LSCs, it is still on the order of $10^{2}-10^{3}$. Further,
without utilisation of hot carrier solar cells coupled to LSCs, Auger
recombination in standard photovoltaics will play a significant role at high
photon flux, potentially hampering the positive gain of large concentration
factors [20].

##  V Conclusion

Considering the propagation and transformation of radiation in the
thermodynamic limit, we have obtained the entropy generation rate for a
photon-multiplicative process. From the second law of thermodynamics we have
obtained the concentration limit for an ideal PM-LSC. Unlike traditional LSCs,
the maximum concentration in a PM-LSC is dependent on brightness of the
incident field. For very high concentration factors, particularly in bright
conditions, the PM-LSC will require a large thermal loss, analogous to the
Stokes shift in a traditional LSC. We find at low brightness, the PM-LSC may
exceed the concentration limit of traditional LSCs. Further, at low
brightness, the improvements required in the photon multiplier efficiency need
only be small to have a dramatic impact on the concentration limit.

###### Acknowledgements.

The authors are thankful to the Engineering and Physical Sciences Research
Council (EPSRC) for financial support. T.K.B. recognises the support of the
Centre for Doctoral Training in New and Sustainable Photovoltaics. A.A
acknowledges funding from the Gates Cambridge Trust and the as well as support
from the Winton Programme for the Physics of Sustainability.

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# Gnevyshev gap in the large-scale magnetic field

V.N.V.N. Obridko 0000-0001-5100-806X  A.S.A.S. Shibalova0000-0002-7228-6226
D.D.D.D. Sokoloff0000-0002-3441-0863  IZMIRAN, 4, Kaluzhskoe Shosse, Troitsk,
Moscow, 142190, Russia  IZMIRAN, 4, Kaluzhskoe Shosse, Troitsk, Moscow,
142190, Russia, Department of Physics, Lomonosov Moscow State University,
Moscow, 119991, Russia and Moscow Center for Fundamental and Applied
Mathematics, Moscow, 119991, Russia

###### Abstract

The phenomenon of the Gnevyshev gap was first identified in the solar-corona
irradiance data (green line). Later, it was studied in the sunspot, coronal,
and heliospheric data. We have investigated the Gnevyshev gap in the magnetic
field data and have arrived at the conclusion that it reflects the behavior of
the large-scale magnetic field. The Gnevyshev gap occurs at the polarity
reversal of the solar magnetic field at the photosphere level. The presence of
the Gnevyshev gap in sunspot data at the photosphere level is disguised by
non-global structures that retain dependence on both latitude and longitude
(the accepted mathematical term is tessaral, see below for more details).
However, it is clearly visible in the magnetic field data at the photosphere
level and is even more pronounced at the source surface (i.e., in the corona).

###### keywords:

Solar Cycle, Observations; Sunspots, Statistics; Solar Irradiance

##  1 Introduction

In 1967, M.Gnevyshev (Gnevyshev, 1967) drew attention to the fact that, in the
activity Cycle 19 (1954–1965), the emission of the solar corona in the
spectral line $\lambda=5303$ Å(green line) had two maxima — in 1957 and in
1959-1960. A decrease in the half-year mean coronal brightness was called the
Gnevyshev gap (GG). This phenomenon has since been observed in various
indices, mainly related to the occurrence rate of solar chromospheric flares,
the upper layers of the solar atmosphere, the interplanetary medium and cosmic
rays. Some authors associated it with quasi-biennial variations of solar
activity (for review, see Storini et al., 2003; Bazilevskaya, Makhmutov, and
Sladkova, 2006; Bazilevskaya et al., 2014 and references therein, Krainev et
al., 2015).

It should be noted that the two-peak structure of the solar cycle is quite
rare when the standard 13-month smoothing is applied (Fig.1). An indication of
their existence can be seen only in 7 cycles (5, 11, 12, 16, 22, 23, and 24)
out of 24 cycles for which reliable data are available (Fig. 1). However, in
cycles 11. 12. 16, generally speaking, there are no dips; there are individual
weak intensifications during the growth or decline phase, and the cycle
remains single-peaked. In cycles 5, 22, 23, and 24, the dip is actually
observed to be 10-15% deep. The peak ratio varies from 1.098 to 1.184, with
the highest secondary maximum observed in cycle 24, and only in cycle 22 is
the secondary maximum lower than the first (see
http://www.solen.info/solar/cycles1_to_present.html or
https://www.sidc.be/SILSO/datafiles).

On a shorter timescale (monthly smoothing), we usually get more than three
peaks unevenly distributed over a wide range of about $12\pm 6$ months. Kane
(2005) suggested calling these structures the Gnevyshev peaks and Gnevyshev
gaps. They can be identified in various photospheric indices and are even more
pronounced in indices related to higher layers of the solar atmosphere
(chromosphere and corona). Kane (2005) analyzed various interplanetary
indices, such as the solar wind density, velocity, and magnetic field, as well
as some geophysical indices, to find out that not a single peak obtained by
monthly averaging could be identified for the entire set of indices under
investigation.

Figure 1:  The dependence of sunspot numbers in Cycles 1-24 (divided into four
panels) on time (years). The monthly mean numbers (vertical axis) are given in
black; red shows the standard 13-month averaging. Time in years is given in
horizontal axis. (Sunspot data from the World Data Center SILSO, Royal
Observatory of Belgium, Brussels. Version V2)

On the other hand, in the cosmic ray and interplanetary data, one can clearly
see a time interval when the solar activity decreases. For this period we
retain the name Gnevyshev gap (GG), although this effect may not actually be
observed in the photosphere.

Thus, the Gnevyshev gap concept looks different in different solar activity
tracers, and the overall situation looks quite complicated. It seems (cf.
Storini et al., 2003) as if there is a filter that prevents the GG-effect from
propagating to higher levels of the solar atmosphere and into the
interplanetary medium. On the other hand, Bravo, Stewart, and Blanco-Cano
(1998) showed that the dipole-like component, which becomes minimal during GG,
is not dominant in the photosphere, but its role increases in the corona.

In our opinion, the GG concept needs clarification and a deeper physical
interpretation. Bravo, Stewart, and Blanco-Cano (1998) pointed out a very
important fact. So far, GG has been studied using sunspot data or data from
high levels of the solar atmosphere or even the interplanetary medium, while
the role of the global magnetic field has been outside our scope of attention.
We believe that the data used so far without taking into account global fields
are insufficient to understand the nature of GG. Indeed, it is reasonable to
believe that GG may be associated with processes that happens in the solar
interior or just at the solar surface. Therefore the surface data should be
most informative for understanding the GG phenomenon (unfortunately, we can
not observe the magnetic field in the solar interior directly). Sunspots are
helpful in this regard; however, they occupy only a tiny part of the solar
surface — no more than a few thousandths even during the solar maximum. As for
the magnetic flux, the sunspot contribution is also much smaller than the
integral solar magnetic flux (Obridko et al., 2022).

Here, it is natural to proceed from the well-known paper by Wang and Sheeley
(2003), who described the GG phenomenon in terms of the magnetic-field
transport by emerging sunspots. We appreciate the scenario presented therein,
but our purpose is to expand the observational basis and involve in
consideration source-surface magnetic field.

We believe that to understand the nature of the Gnevyshev Gap it is necessary
to involve data of the surface large-scale magnetic field, and this is the
goal of our article. In particular, we consider the energy indices of the
solar surface magnetic field to study their cyclic variations and clarify the
nature of the GG phenomenon.

##  2 Gnevyshev gap and energy indices of the solar large-scale magnetic field

Let us apply the method of spherical analysis to estimate the contribution of
different structural parameters to the cyclic evolution of large-scale fields.
We shall use photospheric magnetic field data obtained at Stanford with 3 arc
min resolution for the period of June 1976 to July 2022, i.e., for Carrington
rotations 1642—2055 obtained from http://wso.stanford.edu/forms/prsyn.html. We
assume that in a spherical layer from the photosphere surface of radius
$R_{0}$ to a fixed spherical surface of radius $R_{s}$, conventionally called
the source surface, the magnetic field is completely described by the
potential approximation PFSS (Hoeksema, 1984, 1991). This yields the following
set of magnetic field components $B_{r},B_{\theta},B_{\phi}$ in a spherical
coordinate system $r,\theta,\phi$ (polar radius, latitude, and longitude,
respectively):

$\displaystyle
B_{r}=\sum_{l,m}P_{l}^{m}(\cos\theta)(g_{i}^{m}\cos\phi+h_{i}^{m}\sin
m\phi)\times$ (1)
$\displaystyle\times((l+1)(R_{0}/r)^{l+1}-l(r/R_{s})^{l+1}c_{l}\,,$
$\displaystyle B_{\theta}=-\sum_{l,mm}\frac{\partial
P_{i}^{m}(\cos\theta)}{\partial\theta}(g_{l}^{m}\cos m\phi+h_{l}^{m}\sin
m\phi)\times$ (2)
$\displaystyle\times((R_{0}/r)^{l+2}+(r/R_{s})^{l-1}c_{l})\,,$ $\displaystyle
B_{\phi}=-\sum_{l,m}\frac{m}{\sin\theta}P_{l}^{m}(\cos\theta)(h_{i}^{m}\cos
m\phi-g_{l}^{m}\sin m\phi)\times$ (3)
$\displaystyle\times((R_{0}/r)^{l+2}+(r/R_{s})^{l-1}c_{l})\,.$

Here, $0\leq m$, $l<N$ (usually, $N\leq 9$), $c_{l}=-(R_{0}/R_{s})^{l+2}$,
$P_{l}^{m}$ are the Legendre polynomials, and $g_{l}^{m}$, $h_{l}^{m}$ are the
harmonic coefficients. The latter was calculated from WSO Stanford data. To
find the harmonic coefficients, $g_{l}^{m}$ and $h_{l}^{m}$, and thus, to
fully determine the solution, we had to use boundary conditions. The radial
component of the magnetic field on the photosphere surface plays the role of
one boundary condition, while the other is the assumption that the magnetic
field at the source surface is purely radial.

In order to estimate the magnetic energy at the sphere of radius $r$ we
introduce as an index the mean value of $B_{r}^{2}$ taken at the sphere of
radius $r$ (Obridko and Yermakov, 1989; Obridko and Shelting, 1992):

$i(B_{r})_{|r}=<B_{r}>_{|r}\,,$ (4)

which yields

$i(B_{r})_{|R_{0}}=\sum_{l,m}\frac{l+1+l\zeta^{2l+1})^{2}}{2l+1}((g_{l}^{m})^{2}+(h_{l}^{m})2)$
(5)

and

$i{B_{r}}_{|R_{s}}=\sum_{l,m}(2l+1)\zeta^{2l+4}((g_{l}^{m})^{2}+(h_{l}^{m})2)$
(6)

for the surface and source radii, respectively. Here, $\zeta=R_{0}/R_{s}$. In
our calculations, the radius of the source surface is $R_{s}=2.5R_{0}$ and,
thus, $\zeta=0.4$.

Fig. 2 illustrates the evolution of $i(B_{r})$ calculated at the photoshere
for the time interval 1976–2022, when the required observational data are
available. The two-peak structure is well-pronounced in all four cycles
presented. The later peak is always higher than the earlier one. The
quantitative characteristics of the two-peak structure and the Gnevyshev gap
are given in Table 1.

Figure 2: Evolution of $i(B_{r})$ at the photoshere for 1976–2022 measured in $\mu T^{2}=10^{-4}$ G2 (upper panel, black points show the values for each Carrington rotation, blue points show the data smoothed over 13 rotations) compared to sunspot data smoothed over 13 months (lower panel, cycle numbers are indicated, SSN are given on the right). The vertical red lines show the dates of magnetic field reversal at the North pole; the blue lines stand for the reversals at the South pole. The times of the polar reversals were taken from http://wso.stanford.edu/Polar.html. Table 1: Gnevyshev gap parameters in the last four cycles according to the surface mean magnetic field data. $N_{1}$, $T_{2}$ are the dates of the first and second maxima respectively, $h_{1}$, $h_{2}$ are their heights, and $h_{GG}$ is the height of the Gnevyshev gap. Cycle number |  $T_{1}$, yrs |  $T_{2}$, yrs |  $T_{2}-T_{1}$, yrs | $h_{2}/h_{1}$ | $h_{GG}/h_{1}$
---|---|---|---|---|---
21 | 1979.3 | 1981.3 | 2.0 | 1.25 | 0.848
22 | 1989.4 | 1991.3 | 1.9 | 1.23 | 0.73
23 | 1999.9 | 2002.4 | 2.5 | 1.74 | 0.726
24 | 2012.2 | 2015.0 | 2.8 | 1.69 | 0.778

As a matter of fact, the phenomenon of the Gnevyshev gap actually consists of
two related, but not identical processes (cf. Kane, 2005) — a moderate
decrease in the cycle amplitude after the first maximum and the development of
a secondary maximum. This becomes clearer if we compare the energy indices at
the photosphere with those at the source surface, which are associated with
magnetic fields of larger scale, Fig. 3. The relative contribution of both
phenomena depends on the level at which they are considered.

As seen in Fig. 3, the Gnevyshev gap is more pronounced at the source surface
than at the photosphere. More precisely, the first peak at the source surface
is significantly lower than at the photosphere, and the second one is
significantly higher.

Figure 3: Evolution of the energy index calculated at the photosphere level
(upper panel) compared to that calculated at the source surface (middle
panel), measured in $\mu T^{2}=10^{-4}$ G2 . The lower panel represents the
variation of sunspot numbers.The black dots indicate the values for each
Carrington rotation, the blue line shows the smoothing over 13 rotations.

##  3 Magnetic field symmetries and the structure of the Gnevyshev gap.

The processes associated with the GG phenomenon can be described in more
detail by considering the contribution of magnetic fields of different
symmetries.

The large-scale magnetic fields can be divided by harmonic coefficients into
the following structures of different symmetries (Obridko and Yermakov, 1989;
Shelting, Obridko, and Yermakov, 1989; Obridko and Shelting, 1992):

  * •

Zonal structures ($m=0$). Harmonics are called zonal if their values change
only with latitude. Within one zone, separated from others from the north and
south by parallels, the zonal harmonic retains its sign. These, in turn, can
be divided into zonal odd – ZO ($l$ is odd, the most important case is the
axial dipole, $l$=1) and zonal even – ZE ($l$ is even, the most important case
is the axial quadrupole, $l$=2);

  * •

Sectorial structures $m=l$. Harmonics are called sectorial if their values
change only with longitude. Within one sector, separated from others by
meridians, the sectorial harmonic retains its sign. These, in turn, can be
divided into sectorial odd – SO ($l$ is odd, the most important cases are the
equatorial dipole and two-sector structure with $l=m=1$) and sectorial even –
SE ($l$ is even, say $l=m=2$);

  * •

Tesseral structures ($m>l$); i.e., all other harmonics. These harmonics are so
named because the curves on which they vanish are l-m parallels of latitude
and 2m meridians, which divide the surface of a sphere into quadrangles whose
angles are right angles (tessaras) (Whittaker and Watson, 1990).

Of course, for each type of symmetry it is possible to introduce partial
energy indices, the sum of which will be equal to the general indices
discussed above. We performed a such calculation and normalized the partial
indices to the general ones in order to exclude the general drop of the solar
activity during the last solar cycles (Fig. 4 left panel and Fig. 4 right
panel for the photosphere level and the source surface, respectively).

Figure 4: Evolution of energy indices at the photoshere (top) and at the
source surface (bottom). The upper row shows the contribution of the zonal
harmonics (measured in $\mu T^{2}=10^{-4}$ G2) , the middle row is the
contribution of the sectorial harmonics, and the lower row corresponds to the
tesseral harmonics. The indices are normalized to the total magnetic energy at
the level under consideration.

Comparing Fig. 3 and Fig. 4 we see that the GG in the photoshere is associated
with a substantial drop of the zonal harmonics. At the photosphere level the
drop is slightly compensated since the tesseral harmonics associated with
intermediate and small space doesn’t change very much. The second peak is
associated with the increase in sectorial harmonics, which occurs later. The
GG is more pronounced at the source surface where the relative contribution of
sectorial harmonics is much larger than that of the zonal and tesseral ones.
This explains why the GG phenomenon associated mainly with the sectorial
harmonics was first isolated and then studied using coronal and heliospheric
data.

To summarize the above, we can conclude that the GG phenomenon is associated
with the evolution of the large-scale magnetic field and arises due to the
fact that different field components have a phase shift relative to each
other. The gap occurs because the contribution of the zonal (i.e.,
axisymmetric) harmonic, especially its odd part (ZO), decreases substantially.
Later, at the end of the maximum phase, the contribution of the zonal even
(ZE) and the sectorial harmonics increases. This is why the sectorial
components of the interplanetary magnetic field are most pronounced at the
beginning of the decay phase. Remarkably, the GG is most clearly seen in the
indices where the contribution of the large-scale magnetic field is best
pronounced.

It is significant that the Z0 component associated with the rotation of the
magnetic dipole is most pronounced during the reversals of the polar magnetic
field. Note that the field reversals in the opposite solar hemispheres are not
strictly simultaneous. So, there are time intervals when the magnetic field at
both poles has the same sign. Such periods were the longest in Cycles 22 and
24 (see Fig.2 and Table 1).

##  4 Internal structure of the Gnevyshev gap

More information about the internal structure of the Gnevyshev gap can be
obtained from large-scale magnetic field data. Let us analyze them using Cycle
22 as an example. A year and a half before the maximum (18 June 1988), we see
at the source surface a standard picture dominated by the zonal structure. The
north hemisphere is almost entirely occupied by the field of negative polarity
and the south hemisphere, by the field of positive polarity. The situation
changes dramatically in half a year (26 December 1988): the positive polarity
penetrates the north hemisphere while the negative polarity penetrates the
south one. Even more changes are seen for 08 June 1989: the positive polarity
has approached the north pole. At the end of this year, the positive polarity
reaches the north pole, which can be regarded as the polarity reversal.
However, the positive polarity does not cover the whole north hemisphere,
while the negative polarity has not yet reached the south pole. At this time,
the sunspot number in the photosphere becomes maximum, which by convention is
indicative of the the solar maximum. However, this second maximum in the
sunspot data is lower than the first one, while in the photospheric magnetic
field data, the second maximum is higher than the first one (see Fig. 1) (see
Tabl. 1).

Later, in the first half of 1990, the sunspot number and the magnetic field
decrease. This is precisely what is called the Gnevyshev gap. The zonal odd
harmonics are virtually absent, and the field structure is similar to the
equatorial dipole or quadrupole. The positive polarity dominates at the north
pole until the next field reversal in Cycle 23 when the polar magnetic field
(see Fig. 5) changes sign. In contrast, both polarities are visible near the
south pole. This situation without a distinct prevalence of one or the other
polarity lasts almost 2 years, and the field reversal in the south hemisphere
takes place only in April 1991. At that time, the magnetic field increases
substantially giving rise to the second maximum, which is most pronounced in
the evolution of the source-surface magnetic field. The standard structure
when each hemisphere is dominated by a single polarity and the magnetic field
is determined by the zonal odd harmonic is only restored in the middle of
1992.

Figure 5: Evolution of the polar magnetic field at the north (red curve) and
south (blue curve) poles. The vertical line marks the minimum of the Gnevyshev
gap.

To summarize, we can say that, the Gnevyshev Gap does not manifest amplitude
variations in one or several magnetic-field tracers, but demonstrates a rather
complex spatial interaction of its different components.

##  5 Conclusion

The above results lead us to the conclusion that the Gnevyshev gap reflects
the behavior of the large-scale magnetic field. It occurs at polarity
reversals of the magnetic field at the photosphere level. The presence of GG
in sunspot data is disguised by non-global (mainly tesseral) structures;
however, it becomes clearly visible in the magnetic data in the photosphere
and, especially, at the source surface (i.e., in the corona). That is why the
GG was first identified in coronal observations and was later investigated
using the indices associated with the magnetic field structure in the upper
layers of the solar atmosphere or in the heliosphere. In general, the old
dispute about the nature of the Gnevyshev gap (see Storini et al., 2003;
Bazilevskaya, Makhmutov, and Sladkova, 2006; Bazilevskaya et al., 2014 and
references therein, Krainev et al., 2015) seems to be clarified.

Regarding the theoretical interpretation of GG by Wang and Sheeley (2003), we
appreciate the scenario presented therein; however, we emphasize that the
entire description of the phenomenon must explicitly include a large-scale
magnetic field component. The theoretical ideas of Wang and Sheeley (2003) are
focused on the large-scale magnetic field and we present the observational
basis of the GG using mean-field data.

According to the original definition (Gnevyshev, 1967), the GG phenomenon is
commonly understood as a significant dip near the cycle maximum on the curve
of sunspot numbers smoothed by the standard procedure (13-month smoothing)
with two significant peaks before and after it. Smaller peaks and valleys can
be observed in the monthly mean curve of sunspot numbers during the cycle in
all its phases. They can be caused by the emergence of one or several active
regions or, conversely, by their absence and are not identified as GG
phenomena.

The gap is closely related to the phenomenon of reversal of the polar magnetic
field. Of course, the polar magnetic-field reversal itself is not the initial
cause of the gap, but rather serves as an indicator. Therefore, the gap can be
observed somewhat later than the polarity reversal, as was the case in the
Cycle 23. The immediate physical mechanism of the gap occurrence is the
changing relationship between the components of the large-scale magnetic
field: a sharp decrease in the zonal and increase in the sectorial component,
which leads to discontinuity and the subsequent peak. On the photosphere
surface, the gap is less pronounced, since the fields of smaller scales depend
weakly on the cycle. In the corona, the contribution of small scales drops,
and, therefore, the GG phenomenon is seen much more clearly. Following Wang
and Sheeley (2003) we stress that the polar magnetic field reversal happens
together with the growth of equatorial dipole component which plays a key role
in the GG formation at least during the recent cycles.

The authors thank Todd Hoeksema for presentation of the data.

All authors contributed to the study conception and design. Material
preparation, data collection and analysis were performed by ASS under the
guidance of VNO and DDS. The first draft of the manuscript was written by VNO
and all authors commented on previous versions of the manuscript. All authors
read and approved the final manuscript.

VNO and DDS acknowledge the support of the Ministry of Science and Higher
Education of the Russian Federation under the grant 075-15-2020-780 (VNO) and
075-15-2022-284 (DDS). DDS thanks support by BASIS fund number 21-1-1-4-1 and
the financial support of the Ministry of Education and Science of the Russian
Federation as part of the program of the Moscow Center for Fundamental and
Applied Mathematics under the agreement № 075-15-2022-284.

All data on photospheric magnetic fields are available at the
http://wso.stanford.edu/synopticl.html And Sunspot data from the World Data
Center SILSO, Royal Observatory of Belgium, Brussels. Version V2.

The authors declare that they have no conflict of interest.

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