# Theory of magnetic excitations in multilayer nickelate superconductor
La3Ni2O7

Steffen Bötzel  Theoretische Physik III, Ruhr-Universität Bochum, D-44780
Bochum, Germany  Frank Lechermann  Theoretische Physik III, Ruhr-Universität
Bochum, D-44780 Bochum, Germany  Jannik Gondolf  Theoretische Physik III,
Ruhr-Universität Bochum, D-44780 Bochum, Germany  Ilya M. Eremin  Theoretische
Physik III, Ruhr-Universität Bochum, D-44780 Bochum, Germany

###### Abstract

Motivated by the recent reports of high-$T_{c}$ superconductivity in La3Ni2O7
under pressure, we analyzed theoretically the magnetic excitations in the
normal and the superconducting state in this compound, which can be measured
by inelastic neutron scattering or RIXS. We show that the bilayer structure of
the spin response allows to elucidate the role of the interlayer interaction
and the nature of the Cooper-pairing in a very efficient way. In particular,
we demonstrate the key difference between the potential $s_{\pm}$ and $d$-wave
gaps, proposed recently, by comparing the corresponding response in the even
and odd channels of the spin susceptibility. We show that the mostly
interlayer driven bonding-antibonding $s_{\pm}$ Cooper-pairing produces a
single large spin resonance peak in the odd channel only near the $X$ point
whereas spin resonances in both the odd and the even channel are predicted for
the $d$-wave scenario.

Introduction.— The discovery of unconventional superconductivity in hole-doped
thin films of infinite-layer and reduced multilayer nickelates [1, 2, 3] has
stimulated further interest in studying exotic quantum states and potential
superconductivity in the so-called Ruddlesden-Popper (RP) series of the
nickel-based oxides, denoted as Rn+1NinO3n+1, where R refers to rare earth
element and $n$ is the number of consecutive layers. The most recent
breakthrough in this direction are reports of high-pressure superconductivity
around 80 K in La3Ni2O7 [4, 5, 6, 7, 8, 9, 10, 11] and around 20 K in
La4Ni3O10 [12, 13, 14, 15]. These exciting discoveries motivated massive
theoretical investigation [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 22, 45, 46,
47, 48, 49, 50, 51, 52], yet the full structural characterization of these
systems is still under debate [53, 54, 7]. Nevertheless, a striking increase
of the superconducting transition temperature in La3Ni2O7 with multi- or
bilayer structure calls for a careful theoretical examination.

Considering La3Ni2O7 as RP bilayer systems yields a formal Ni $3d^{7.5}$ (or
$3d^{8}$ when considering ligand-hole physics [16]) electronic configuration
with both Ni-$e_{g}$ orbitals crossing the Fermi level. The low energy physics
in this system is ruled by the multiorbital and the bilayer effects with the
strong hybridization between the Ni-$d_{z^{2}}$ and the apical O-$p_{z}$
orbitals. The multiorbital structure seems to be one of the key differences
between La3Ni2O7 and the bilayer cuprate superconductors where Cu2+ ions with
$3d^{9}$ configuration possess only one unpaired valence electron in the
$3d_{x^{2}-y^{2}}$-orbital, whereas the Ni ion has unpaired valence electrons
in both the $3d_{x^{2}-y^{2}}$ and $3d_{z^{2}}$ orbitals. Various Hubbard-
Hund-type or $t-J$ like models have already been proposed to capture the
superconducting and normal state properties of this multiorbital system [16,
17, 20, 19, 21, 18, 22, 29, 30, 31, 32, 33, 34, 28, 35, 36, 37]. Within the
variety of model considerations, one of the most interesting theoretical
question concerns the interplay between the intralayer and the interlayer
Cooper pairing, which yields a competition between the $s_{\pm}$-wave symmetry
of the superconducting order parameter, driven mostly by the interlayer
Cooper-pairing [17, 16, 20, 23, 24, 19, 18, 22, 25, 21, 27, 26] and the
$d_{x^{2}-y^{2}}$-wave or the $d_{xy}$-wave symmetries of the superconducting
order parameters, driven mostly by the intralayer interaction, respectively
[16, 17, 28, 27, 45].

Given the likely non-phononic origin of superconductivity in pressurized
La3Ni2O7, the strange metal behavior of the normal state [6, 4, 10] and the
signatures of magnetic ordering, seen at the ambient pressure [55, 56] at
around 150 K, it is instructive to study the bilayer-structure impact on the
spin response in this system in the normal and superconducting states. Recall
that one of the important hallmarks of the superconducting state in
unconventional superconductors is the occurrence of the so-called spin
resonance peak. It is seen by the inelastic neutron scattering in various
systems at the antiferromagnetic (AFM) wave vector $\mathbf{Q}_{\rm
AFM}=(\pi,\pi)$ at energies below or around the superconducting gap energy
threshold of about 2$\Delta$ [57]. Its presence in various unconventional
superconductors ranging from high-Tc cuprates [58, 59, 60, 61, 62], iron-based
superconductors [63, 64, 65, 66], and some heavy-fermion superconductors like
CeCoIn5 [67, 68] is considered to be one of the strong signatures of spin-
fluctuation mediated Cooper-pairing [57, 69]. In the simplest theoretical
picture, the spin resonance peak occurs due to a change of sign of the
superconducting order parameter at the parts of the Fermi surface which are
connected by the AFM wave vector $\Delta_{\mathbf{k}}$ and
$\Delta_{\mathbf{k}+\mathbf{Q}}$ [70, 71, 72, 73, 74, 75, 76, 77, 78].

The occurrence of the resonance peaks and their dispersion in bilayer systems
not only allows to confirm the unconventional nature of superconductivity and
to learn about the superconducting gap symmetry, but also to understand the
importance of the interlayer coupling. In particular, due to two CuO2 layers
per unit cell, the spin response in bilayer cuprates splits into even and odd
channels. This in turn can be connected to the bonding and anti-bonding
character of the electronic bands, showing modulations in $q_{z}$ directions
[79]. The magnetic susceptibility splits into even $\chi_{e}$ and odd
$\chi_{o}$ susceptibilities [80, 81, 82, 83] and they can be accessed
individually by measuring the spin response at different $q_{z}$ momenta. In
bilayer cuprates, the spin resonance peak was found in both channels in a wide
doping range with clear splitting between $\chi_{e}$ and $\chi_{o}$ [84, 85].
Furthermore, analyzing their intensities and frequency positions, one was able
to extract important information of the overall structure of the paramagnon
continuum and the interplay of interlayer exchange interactions and interlayer
hopping matrix elements [80, 81, 82, 83]. In La3Ni2O7 the situation is even
more intriguing, because the splitting of bonding and anti-bonding bands is
much stronger.

In this work, we provide a theoretical description of the spin response of
paramagnetic La3Ni2O7 in the pressurized normal and superconducting state. We
begin by considering the spin response in multiorbital bilayer systems and
then focus on two different scenarios for the superconducting pairing. First,
we consider $s_{\pm}$ interlayer pairing with a sign change between bonding
and anti-bonding bands. Using the tight-binding model from Ref. [33] we
discuss the important scattering vectors contributing to $\chi_{e}$ and
$\chi_{o}$. We find that $\chi_{o}$ gives a much larger spin response. In the
superconducting state, this scenario manifests in a dominant spin resonance
peak near the X-point. The second scenario is a cuprate like $d_{x^{2}-y^{2}}$
gap symmetry which is discussed in the context of the model presented in Ref.
[16]. Here, the spin response for $\chi_{e}$ and $\chi_{o}$ channels is of the
same order. Correspondingly, this scenario yields different spin resonance
peaks in the superconducting state in both $\chi_{e}$ and $\chi_{o}$.
Measuring the spin response in both channels can therefore clearly reveal the
actual Cooper-pairing scenario and clarify the role of the interlayer exchange
interaction in this system.

Theoretical Approach.— We consider an effective quasi-two-dimensional bilayer
Hamiltonian for the Ni-$e_{g}$ orbitals, illustrated in Fig. 1(a), which
follows from the tight-binding models developed previously [33, 16]. The
Hamiltonian reads

$\displaystyle\mathcal{H}$
$\displaystyle=\mathcal{H}_{0}+\mathcal{H}_{\text{on-
site}}+\mathcal{H}_{\perp},$ (1)

where $\mathcal{H}_{\text{on-site}}$ and $\mathcal{H}_{\perp}$ denote on-site
and interlayer interactions between Ni atoms within a bilayer sandwich via the
apical oxygen, respectively.

Figure 1:  (a) Schematic representation of the bilayer model with $e_{g}$
orbitals. (b) and (d) show the Fermi surface for the model [33] with bonding
$(b)$ and anti-bonding $(a)$ bands shown by blue (solid) and red (dashed)
curves, respectively. Note the sign-changing bonding-antibonding $s_{\pm}$-gap
follows the same red-blue color distribution referring to the negative and
positive regions of the gap. Greek indices denote the notion of the bands,
accepted in literature. The arrows in (b) and (d) highlight the important
scattering wavevectors in the even and odd channels, respectively. (c) and (e)
show the even ($\chi_{e}$) and the odd ($\chi_{o}$) parts of the static RPA
susceptibility, respectively, calculated in the normal state ($T=80$ K) for
$U=0.9U_{\text{mag}}$ and $J_{\text{H}}=U/7$. Peaks are labeled according to
scattering vectors displayed in (b) and (d).

For the non-interacting part $\mathcal{H}_{0}$, we also restrict ourselves to
the interlayer hopping within the bilayer sandwiches for both $d_{z^{2}}$ and
$d_{x^{2}-y^{2}}$ orbitals [33, 16]. In momentum space it reads

$\mathcal{H}_{0}=\sum_{\mathbf{k},k_{z}}\sum_{l_{1},l_{2}}\sum_{o_{1},o_{2}}\hat{H}_{l_{1}o_{1};l_{2}o_{2}}(\mathbf{k},k_{z})c^{\dagger}_{l_{1}o_{1}}(\mathbf{k},k_{z})c_{l_{2}o_{2}}(\mathbf{k},k_{z}),$
(2)

where $c^{\dagger}_{l,o}(\mathbf{k},k_{z})$ creates an electron in layer $l$
and orbital $o$ with in-plane momentum $\mathbf{k}=(k_{x},k_{y})$. The out-of-
plane momentum $k_{z}$ is only needed to carefully describe the Fourier
transform for a bilayer system and the upper and the lower layer are denoted
as $A$ and $B$, respectively. As the Hamiltonian must be invariant under
exchange of layers, we find the general form

$\hat{H}(\mathbf{k},k_{z})=\begin{pmatrix}\hat{H}_{AA}(\mathbf{k})&\hat{H}_{AB}(\mathbf{k})e^{i\phi}\\\
\hat{H}_{AB}(\mathbf{k})e^{-i\phi}&\hat{H}_{AA}(\mathbf{k})\end{pmatrix},$ (3)

where $\hat{H}_{AA}$ and $\hat{H}_{AB}$ incorporate the intralayer and
interlayer hoppings, respectively. Here the $k_{z}$ dependence only appears
due to a phase factor $\phi=-k_{z}d$ with $d$ being the height of the bilayer
sandwich. The hats are put to remind the reader that all blocks are, in
principle, square matrices in the orbital degree of freedom. The above
Hamiltonian can be block-diagonalized using the transformation

$\displaystyle
V=\frac{1}{\sqrt{2}}\begin{pmatrix}\mathbb{1}&\mathbb{1}e^{i\phi}\\\
\mathbb{1}e^{-i\phi}&-\mathbb{1}\end{pmatrix},\ \ \ \
\hat{H}^{b/a}=\hat{H}_{AA}\pm\hat{H}_{AB},$ (4)

where the blocks belong to the bonding (b) and anti-bonding (a) sub-spaces.
Note that the phase factor is not present in the (ab)-space.

For the interaction part of the Hamiltonian we include the on-site
intraorbital ($U$) and interorbital ($U^{\prime}$), Hund’s type
($J_{\text{H}}$) and pair hopping ($J^{\prime}$) interaction and assume spin-
rotational invariance, which yields the relations $U^{\prime}=U-2J_{\text{H}}$
and $J_{\text{H}}=J^{\prime}$ [16, 86]. In addition to the hybridization of
the layers due to hopping matrix elements in the non-interacting part of the
Hamiltonian, we also include the interlayer superexchange interaction between
the neighboring 3$d_{z^{2}}$-orbitals via oxygen $p_{z}$-orbital

$\displaystyle\mathcal{H}_{\perp}$
$\displaystyle=J_{\perp}\sum_{i}\left(\mathbf{S}_{A,i}\mathbf{S}_{B,i}-\frac{1}{4}n_{A,i}n_{B,i}\right)$
$\displaystyle=J_{\perp}\left[\mathbf{S}_{A}(\mathbf{q}_{)}\mathbf{S}_{B}(-\mathbf{q})-\frac{1}{4}n_{A}(\mathbf{q})n_{B}(-\mathbf{q})\right]e^{i\phi}.$
(5)

Here $\mathbf{S}_{A,i}$ and $n_{A,i}$ are the spin and charge density
operators at lattice site $i$ and sublattice A, respectively. Note the
occurrence of the phase factor $e^{i\phi}$ for the interlayer part of the
Hamiltonian, which is absent for the on-site part. Throughout this work, we
also neglect the interlayer superexchange interaction for the
Ni-$d_{x^{2}-y^{2}}$ orbitals, which is expected to be much weaker than it is
for the Ni-$d_{z^{2}}$-orbitals. Note, in the bilayer cuprates the difference
of the spin response in the odd and the even channel can be simply explained
by the effect of the interlayer hopping matrix elements without invoking the
interlayer superexchange interaction [82]

The spin susceptibility in the superconducting state can be written in terms
of normal and anomalous Green’s functions

$\displaystyle\chi_{\text{spin}}^{\alpha\beta}(q)$
$\displaystyle=S^{\alpha}_{\eta_{4}\eta_{1}}S^{\beta}_{\eta_{2}\eta_{3}}\frac{T}{N}\sum_{k}[F_{\eta_{1}\eta_{3}}(k+q)\bar{F}_{\eta_{2}\eta_{4}}(k)$
$\displaystyle-G_{\eta_{1}\eta_{2}}(k+q)G_{\eta_{3}\eta_{4}}(k)],$ (6)
$\displaystyle S^{\alpha}_{\eta_{1}\eta_{2}}$
$\displaystyle=\frac{\hbar}{2}{\sigma}^{\alpha}_{s_{1}s_{2}}\delta_{o_{1}o_{2}}\delta_{l_{1}l_{2}}$
(7)

where we use the four notation $q=(\mathbf{q},i\omega_{n})$ and shorthand
indices $\eta=(l,o,s)$ with implied summation over repeated indices. To
compute the spin susceptibility, we use the random phase approximation (RPA)
[16, 86]. The interacting susceptibility can be written as a matrix equation
with susceptibility and interaction matrices of the form

$\displaystyle\hat{\chi}(\mathbf{q})$
$\displaystyle=\begin{pmatrix}\hat{\chi}_{AA}(\mathbf{q})&\hat{\chi}_{AB}(\mathbf{q})e^{i\phi}\\\
\hat{\chi}_{AB}(\mathbf{q})e^{-i\phi}&\hat{\chi}_{AA}(\mathbf{q})\end{pmatrix},$
(8) $\displaystyle\hat{U}$
$\displaystyle=\begin{pmatrix}\hat{U}_{AA}(\mathbf{q})&\hat{U}_{AB}(\mathbf{q})e^{i\phi}\\\
\hat{U}_{AB}(\mathbf{q})e^{-i\phi}&\hat{U}_{AA}(\mathbf{q})\end{pmatrix}.$ (9)

The Dyson-type RPA matrix equation can be decomposed into even and odd
channels with respect to the exchange of layer index by defining
$\hat{\chi}^{e/o}=2(\hat{\chi}_{AA}\pm\hat{\chi}_{AB})$ and
$\hat{U}^{e/o}=\hat{U}_{AA}\pm\hat{U}_{AB}$:

$\hat{\chi}^{(e/o)}=\left[\mathbb{1}-\frac{1}{2}\hat{\chi}^{(e/o)}_{0}\hat{U}^{(e/o)}\right]^{-1}\hat{\chi}^{(e/o)}_{0},$
(10)

where $\chi_{0}$ denotes the non-interacting susceptibilities. The above
expression holds in general for both, the spin and the charge
susceptibilities, but the interaction matrix $\hat{U}^{(e/o)}$ has to be
chosen differently. The physical paramagnetic spin susceptibility is given in
terms of $\hat{\chi}_{e}$ and $\hat{\chi}_{o}$ by

$\displaystyle\chi_{\text{spin}}$
$\displaystyle=\sum_{l_{1}l_{2},o_{1}o_{2}}\begin{pmatrix}\hat{\chi}^{e}+\hat{\chi}^{o}&(\hat{\chi}^{e}-\chi^{o})e^{i\phi}\\\
(\hat{\chi}^{e}-\hat{\chi}^{o})e^{-i\phi}&\hat{\chi}^{e}+\hat{\chi}^{o}\end{pmatrix}_{l_{1}o_{1};l_{2}o_{2}}$
$\displaystyle=\sum_{o_{1},o_{2}}[\hat{\chi}^{e}_{o_{1},o_{2}}\cos^{2}(\frac{q_{z}d}{2})+\hat{\chi}^{o}_{o_{1},o_{2}}\sin^{2}(\frac{q_{z}d}{2})].$
(11)

This expression of the spin susceptibility has been initially derived for the
bilayer cuprates [79, 73, 82]. By explicitly using the matrix elements from
Eq. 4, we can express the even and the odd susceptibilities through
susceptibilities in the $(ab)$-space:
$\hat{\chi}^{e}_{0}=\hat{\chi}^{aa}+\hat{\chi}^{bb}$ and
$\hat{\chi}^{o}_{0}=\hat{\chi}^{ab}+\hat{\chi}^{ba}$.

Results.— In what follows we compute the bilayer spin response for pressurized
La3Ni2O7 for two slightly different tight-binding parameterizations of the
non-interacting Hamiltonian to observe the general trends. In Fig. 1(c),(e)
and Fig. 3(c) we show the normal state even and odd components of the bilayer
spin susceptibility using the tight-binding model from Ref. [33] and Ref.
[16], respectively. One immediately sees that independent of the model used,
the magnetic response in the odd and the even channels strongly differ from
each other with respect to the dominant scattering peaks. This occurs due to
the large splitting of the bonding and anti-bonding bands of
$3d_{x^{2}-y^{2}}$-like character and the bonding only character of the
$3d_{z^{2}}$-like pocket at the $M$-point, which crosses the Fermi level.
Specifically, the main scatterings in the even channel stem from the
scattering within anti-bonding $3d_{x^{2}-y^{2}}$-band (q${}_{1}^{e}$ and
q${}_{2}^{e}$) as well as scattering between bonding $\alpha$ and
$\gamma$-band (q${}_{3}^{e}$), which is illustrated in Fig. 1(b). At the same
time, the main scatterings in the odd channel come from antibonding to bonding
transition from $\alpha$ to $\beta$ bands (q${}_{2}^{o}$) and $\gamma$ to
$\beta$ bands (q${}_{1}^{o}$), which is displayed in Fig. 1(d). Apart from
different magnitudes, these peaks are dominant in both models. Another feature
we see is that interlayer exchange interaction is not playing a very
significant role as its inclusion does not affect the results. Thus, the
different behavior in both channels arises purely from the bonding-antibonding
splitting of the electronic bands and the exchange interaction, which value we
employ according the ratio $J_{\perp}/U$ from Ref. [38], plays less vital role
in providing difference between odd and even susceptibilities in the bilayer
nickelates as one would naively expect.

Despite the similar q-dependence of the spin response in both models, the
dominant superconducting instabilities appear to be different. While models
based on Ref. [33] and spin-fluctuation mediated pairing mostly predict sign-
changing bonding-antibonding $s$-wave solutions [18, 19], similar spin-
fluctuation based analyses reveal $d$-wave symmetries to be present in the
other model [16, 17]. One can understand this difference by simply looking
into the overall magnitudes of the even and odd susceptibilities. In the case
of the tight-binding model of Ref. [33] the odd susceptibility appears to be
twice larger in magnitude than the even susceptibility, which indicates the
dominance of the interlayer itinerant magnetic fluctuations. This dominance of
the magnetic fluctuations in the odd channel supports the interlayer (bonding-
antibonding) $s_{\pm}$-wave Cooper-pairing with large interlayer
superconducting gap [38, 36], which at the same time do not necessarily
require the strong superexchange interaction between the layers. At the same
time, within the other tight-binding model [16] the different peaks in the odd
and the even susceptibilities appear to have similar magnitudes and among
several candidates, the $d$-wave superconducting gap appears to be the most
stable solution. One should further note at this point that the $d$-wave
symmetry is usually more stable against the inclusion of the local Coulomb
interaction and even in the case of near competition between various channels
this solution usually wins [87].

The strong differentiation of the even and the odd susceptibilities seen in
the normal state offer now a practical tool to disentangle between various
Cooper-pairing scenarios. For this, we now extend the calculations of the spin
response to the superconducting state by comparing the structure of the spin
response for the model giving the sign-changing bonding-antibonding $s$-wave
solution [33] and that, which gives the $d$-wave solution to be the most
dominant [16].

Note, the bonding-antibonding $s_{\pm}$-wave solution in the (ab)-space allows
for a simple decomposition of the Cooper-pairing to the interlayer and
intralayer contribution. In particular, we can write the gaps in the
sublattice space $\hat{\Delta}^{a/b}=\hat{\Delta}\pm\hat{\Delta}^{\perp}$,
where $\hat{\Delta}$ and $\hat{\Delta}^{\perp}$ are the intralayer and the
interlayer superconducting order parameters, respectively. By setting
$\Delta_{x^{2}-y^{2}}^{\perp}=\Delta_{z^{2}}^{\perp}$ and $\hat{\Delta}=0$,
one ends up with a constant magnitude gap, which however changes sign between
bonding and anti-bonding bands. This gap symmetry is consistent with the
numerical solutions found in the literature [17, 16, 20, 23, 24, 19, 18, 22,
25, 21, 27, 26] and is illustrated in Fig. 1(b).

Given the peculiar structure of the sign-changing bonding-antibonding gap, the
spin resonance in the spin susceptibility exclusively appears in the odd
channel with the scattering between bonding and antibonding bands but not in
the even channel. In particular, we show in Fig. 2 the calculated frequency
dependence of the imaginary parts of the even, Im$\chi_{e}$, and the odd,
Im$\chi_{o}$, spin susceptibilities at the characteristic wave vectors
$\mathbf{q}_{1}^{o}$, $\mathbf{q}_{2}^{o}$, and $\mathbf{q}_{1}^{e}$
identified in the normal state and employing the gap size of 15 meV, which
yields a plausible gap-to-$T_{c}$ ratio of about 4.3. Observe that the
enhancement occurs only for the odd spin response at $\mathbf{q}_{1}^{o}$ and
$\mathbf{q}_{2}^{o}$ where the superconducting gap changes sign between
bonding and antibonding bands and the exact magnitude depends on the values of
the superconducting gap at the corresponding region. On the contrary, the spin
susceptibility in the even channel at $\mathbf{q}_{1}^{e}$ is suppressed as
generally the scattering wavevectors within bonding or antibonding bands
connect regions of the same gap sign of the superconducting order parameter.
In this regard, the outlined behavior of the spin response, i.e. the
enhancement of the odd spin susceptibility and its absence in the even channel
in the experiment probed by inelastic neutron scattering or RIXS will be a
direct probe for the interlayer Cooper-pairing and bonding-antibonding
character of the $s_{\pm}$-wave gap. Interestingly enough, one should mention
that the local spin probe such as NMR will see an opposite trend due to a
different coherence factor involved [88]. In particular, while at $T=0$ the
dominant contribution to the spin response comes from the Cooper-pairing
breaking term, which is proportional to $1-\frac{\Delta_{\bf
k}\Delta_{k+q}}{|\Delta_{\bf k}||\Delta_{k+q}|}$ coherence factor when
neglecting interband Cooper-pairing contributions and thus is enhanced if the
order parameter at k and k+q changes sign, the main contribution for $T\sim
T_{c}$ comes from the thermal quasiparticles, which acquires
$1+\frac{\Delta_{\bf k}\Delta_{k+q}}{|\Delta_{\bf k}||\Delta_{k+q}|}$
coherence factor. Consequently, the spin-lattice relaxation time, $1/T_{1}$,
will show a Hebel-Slichter peak for the sign-preserving even spin channel and
its suppression in the odd channel.

Note that the obtained results are robust against small variations of the gap
magnitude. The solutions discussed in literature sometimes show nodal regions
appearing in $\Gamma-M$ direction, which could be incorporated by fine-tuning
and inclusion of interorbital gaps. However, we find that such details are
essentially irrelevant for the generic behavior of the spin resonance.

Figure 2:  Calculated frequency dependence of the imaginary part of the even
and the odd spin susceptibility in the bonding-antibonding s±-wave
superconducting (solid curves) and normal (dashed curves) states using the
tight-binding parameters from Ref. [33] and the maximum gap magnitude of 15
meV. The representative wave vectors $\mathbf{q}_{1}^{o}$ (a),
$\mathbf{q}_{2}^{o}$ (b) and $\mathbf{q}_{1}^{e}$ (c) are chosen from Fig. 1.

Let us now turn to the discussion of the spin response for the second of the
models outlined in Ref. [16], which give the $d_{x^{2}-y^{2}}$-wave
superconducting gap symmetry, shown in Fig. 3(a). Here we should mention that
the large hybridization of the $\alpha$ and $\beta$ bands does not allow
straightforward decomposition of the $d_{x^{2}-y^{2}}$-wave solution in the
orbital and sublattice degrees of freedom and should be viewed in the band
space, which has different contributions including those from interorbital and
inter- and intralayer gaps, which do not yield interband gaps.

Figure 3:  (a) Sketch of the $d_{x^{2}-y^{2}}$-wave gap solution for the
tight-binding model [16] where blue and red colors refer to the opposite signs
of the gap magnitudes. (b) shows the even ($\chi_{e}$) and the odd
($\chi_{o}$) parts of the static RPA susceptibility, respectively, calculated
in the normal state ($T=80$ K) for $U=0.9U_{\text{mag}}$ and
$J_{\text{H}}=U/4$. (c) shows the calculated frequency dependence of the
imaginary part at the characteristic maximal scattering wavevectors of the
even and the odd spin susceptibility for the $d_{x^{2}-y^{2}}$-wave
superconducting (solid curves) and normal (dashed curves) states and the
maximum gap magnitude of 15 meV.

In contrast to the bonding-antibonding s±-wave scenario, a sizeable
enhancement of the Im$\chi$ in the $d$-wave superconducting state is seen in
the odd channel at $\mathbf{q}_{1}^{o}$ as well as in the even channel near
$\mathbf{q}_{1}^{e}$ as shown in Fig. 3(c). The differences in the overall
magnitude can be traced back to the angular dependence of the superconducting
gap at the Fermi surface. For example, there is no sign change of the gap on
the $\beta$\- Fermi surface sheet portions for the wavevectors ${\bf
q}_{1}^{e}$ and ${\bf q}_{2}^{e}$, shown as arrows in Fig. 1(b). In fact, it
is the scattering across the $\gamma$-pocket that causes the enhancement and
happens to be peaked at the same momentum transfer. In contrast to the odd
spin resonance in the s${}_{\pm}-$wave scenario, the resonances in the odd and
even channels in the $d_{x^{2}-y^{2}}$-wave scenario are much broader and for
the even channel also strongly dispersing in the momentum space. Most
importantly, within the $d-$wave scenario there is no selectivity between odd
and even channel in the spin response in the superconducting state, in both
channels there is some enhancement.

Let us also mention that in Refs. [16, 17, 27] the $d_{xy}$-wave solution was
also appearing as one of the leading instability for smaller Hund’s coupling.
Similar to $d_{x^{2}-y^{2}}$ case we find a sharp spin resonance peak (not
shown) in the odd channel at $\mathbf{q}_{1}^{o}$ and a broad peak the even
channel centered at $\mathbf{q}_{2}^{e}$ with the dispersion along $\Gamma-X$
direction, while in the $d_{x^{2}-y^{2}}$ one finds the dispersion along
$\Gamma-M$-direction. Therefore $d_{xy}$ and $d_{x^{2}-y^{2}}$ pairing
symmetries can be distinguished by comparing $\Gamma-X$ and $\Gamma-M$
directions for the even channel of the spin susceptibility.

A different Fermi surface topology in which the $\gamma$ pocket sinks below
the Fermi surface has been predicted in Ref. [29], which is in line with the
recent ARPES results for ambient pressure [89]. For the bonding-antibonding
$s_{\pm}$-wave Cooper-pairing scenario, which changes sign between $\alpha$
and $\beta$ bands, a spin resonance would appear exclusively at
$\mathbf{q}_{2}^{o}$. The scattering vector $\mathbf{q}_{2}^{o}$ may also play
an important role in the magnetic ordering at ambient pressure. In the
reciprocal space index corresponding to the pseudo-tetragonal unit cell
(H,K,L), this peak is located at $(0.34,0.34,c/(2d)\approx 2.6)$ close to the
magnetic peak (0.25,0.25,2.5) found in very recent RIXS measurements at
ambient pressure [90]. We believe that the fact that in experiment this peak
was found to be non-dispersive but loosing intensity away from L$=2.5$ towards
lower L (see Fig. 2c in Ref. [90]) is an indicator of being present in the odd
channel exclusively.

Conclusion.– To conclude, we have shown that the interlayer bonding-
antibonding $s_{\pm}$ and $d$-wave pairing scenarios yield clearly
distinguishable bilayer spin responses in the normal and the superconducting
states. The $s_{\pm}$-wave Cooper pairing gives rise to a strong spin
resonance peak in the odd channel of the spin susceptibility, whereas the even
response shows no enhancement. In contrast, for the $d$-wave symmetry spin
resonance-like weaker peaks appear in both, even and odd, channels. Therefore,
studying bilayer structure of the spin response in the La3Ni2O7 may provide a
crucial check for the superconducting gap symmetry.

##  I Acknowledgements

The work is supported by the German Research Foundation within the bilateral
NSFC-DFG Project ER 463/14-1.

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# On the self-similarity of diffracting gaseous detonations and the critical
channel width problem

M. I. Radulescu<EMAIL_ADDRESS>Department of Mechanical Engineering,
University of Ottawa, Ottawa (ON) K1N 6N5 Canada  R. Mével  Center for
Combustion Energy, School of Vehicle and Mobility, State Key Laboratory for
Automotive Safety and Energy, Tsinghua University, Beijing, China  Q. Xiao
National Key Laboratory of Transient Physics, Nanjing University of Science
and Technology, Nanjing, 210094, China  S. Gallier  ArianeGroup, Le Bouchet
Research Center, 91710 Vert le Petit, France

###### Abstract

One strategy for arresting propagating detonation waves in pipes is by
imposing a sudden area enlargement, which provides a rapid lateral divergence
of the gases in the reaction zone and attenuates the leading shock. For
sufficiently small tube diameter, the detonation decays to a deflagration and
the shock decays to negligible strengths. This is known as the critical tube
diameter problem. In the present study, we provide a closed form model to
predict the detonation quenching for 2D channels. Whitham’s geometric shock
dynamics, coupled with a shock evolution law based on shocks sustained by a
constant source obtained by the shock change equations of Radulescu, is shown
to capture the lateral shock dynamics response to the failure wave originating
at the expansion corner. A criterion for successful detonation transmission to
open space is that the lateral strain rate provided by the failure wave not
exceed the critical strain rate of steady curved detonations. Using the
critical lateral strain rate obtained by He and Clavin, a closed form solution
is obtained for the critical channel opening permitting detonation
transmission. The predicted critical channel width is found in very good
agreement with our recent experiments and simulations of diffracting H2/O2/Ar
detonations.

##  I Introduction

When a detonation wave emerges from a tube or channel into an open space, the
sudden expansion of the gases in the reaction zone of the detonation wave
decouples the reaction zone from the shock near the cornerDremin (1999). If
the opening is sufficiently small, the detonation can completely degenerate
into a weak shock followed by a deflagration wave. This so-called critical
tube diameter problem, or critical channel height in 2D, has attracted much
interest since the pioneering work of Zel’dovich, Kogarko, and Simonov (1956),
due to its practical importance in detonation initiation or quenching
applications. For gaseous detonations, a review of the state of the art can be
found in the PhD thesis of Schultz (2000) for work prior to 2000, while a
discussion of the more recent litterature appears in the notable contributions
of Jones  _et al._ (2000); Meredith, Ng, and Lee (2010); Sorin  _et al._
(2009); Khasainov  _et al._ (2005); Nagura  _et al._ (2013); Li  _et al._
(2016); Shi, Uy, and Wen (2020); Gallier  _et al._ (2017); Pintgen and
Shepherd (2009); Yuan  _et al._ (2021); Li  _et al._ (2021). While this list
is not exhaustive, the scope of our present communication is not to provide a
critical review of the vast available litterature on the subject. We note
however that in spite of numerous efforts in modeling the diffraction process,
a predictive model for the critical tube diameter is still lacking Lee (1984);
Arienti (2003); Lee (2008). The prime objective of the present communication
is to formulate such a model.

The lack of a predictive model is due in part to the presence of a cellular
structure of the front of all detonations, which modifies the reaction zone
structure of detonations and its sensitivity to the global expansion during
the diffraction process, as recently shown by Xiao and Radulescu (2020a). For
example, a unique signature of the transverse wave structure of detonations is
the anomalous scaling of the diffraction process between the 2D channel
problem and 3D axisymmetric problem, as first noted by Radulescu (2003) when
reviewing previous data, and subsequently confirmed experimnetally by
Meredith, Ng, and Lee (2010). This anomalous scaling of a tube diameter to
channel width greater than 2 cannot be reconciled with curvature based
criteria for failure and extensions to include non-steady effectsBdzil and
Stewart (1986), and is beleived to be a unique feature of unstable cellular
detonations. Recovering the correct scaling can be deemed the ultimate acid
test for any theoretical model of diffraction for unstable detonations.
Recently, however, Meredith, Ng, and Lee (2010) and Radulescu and Lee (2002)
in porous walled tubes, have shown that only weakly unstable detonations waves
do obey the ideal curvature-based scaling, substantiating Lee’s claim of the
failure mechanism being associated with a critical curvature Lee (1996). This
ideal curvature based scaling was verified numerically by Li  _et al._ (2016)
for weakly unstable detonations. The present study addresses whether a
curvature based prediction of critical transmission is compatible with
experiments in weakly unstable detonations and can quantitatively predict the
critical diffraction conditions.

Recent work by Xiao and Radulescu (2020b) in slowly enlarging channels has
shown that the quasi-steady dynamics of such weakly unstable cellular hydrogen
detonations at low pressure, which are characterized by much longer reaction
zones as compared to induction zones, can be well captured by the predictions
of the ZND model with curvature Xiao and Radulescu (2020b) and neglect of the
cellular structure. It is thus of interest to verify whether the diffraction
process in these same mixtures can be equally well predicted by neglecting the
influence of the cellular structure for these conditions using a critical
curvature criterion, as suggested by Lee (1996). The model we formulate, based
on a maximum permissible curvature, is thus meant for weakly unstable
detonations. This is the first step towards predicting the behavior of more
unstable detonations, as it can serve as a benchmark for more unstable
conditions.

The prediction of the critical conditions for transmission in open space of a
detonation wave requires modeling the dynamics of the diffracting detonations
and the distribution of transverse flow strain rate (i.e., curvature times
flow speed) behind the lead shock. Previous studies have shown that the
reaction zone decouples from the lead shock behind a failure wave propagating
to the axis Dremin (1999); Arienti and Shepherd (2005). The shape of the de-
coupled shock wave was shown to be approximately self-similar by Bartlma and
Schroder (1986), and approximately captured by Whitham’s geometrical shock
dynamics theory for inert shock waves Whitham (1974). Arienti compared the
curvature predicted by the model of Whitham with his computations and found it
under-predicted the wave curvature Arienti (2003). Wescott, Stewart and Bdzil
(henceforth WSB) have extended Whitham’s characteristic rule to under-driven
detonation waves by assuming quasi-steady dynamics and a reaction zone with an
embedded sonic surface, but did not compare their predictions to gaseous
diffracting detonations with sensible shock sensitivity permitting local
extinction Wescott, Stewart, and Bdzil (2004). Recently, we have conducted
detailed numerical simulations and experiments of diffracting detonations in
weakly unstable 2H2+O2+2Ar detonations Mevel, Xiao, and Radulescu (2017). In
this study, we wish to compare the predictions of Whitham and WSB curvature
distribution behind the diffracting detonation waves with our recent
experiments and simulations.

While both Whitham’s and the WSB models will be shown to provide an adequate
approximation for the lateral strain rate distribution required for predicting
failure, we also introduce a novel self-similar approximation for the shock
dynamics weakly supported by the motion of the detonation products. This model
provides an improved prediction for the detonation dynamics. Combining the
prediction of lateral strain rate and the maximum strain rate obtained for
steady waves He and Clavin (1994); Klein, Krok, and Shepherd (1995); Yao and
Stewart (1995), we obtain a closed form expression for the critical channel
height for detonation transmission. We show that the closed form prediction
for detonation failure is in excellent agreement with the simulations and good
agreement with experiment.

The paper is organized as follows. We first briefly review our previous
numerical results and experiments. We then compare the detonation dynamics
with the predictions of Whitham’s Geometrical Shock Dynamics using the Whitham
and Westcott’s truncations utilizing the characteristic rule, as well as our
weakly supported shock model. We conclude by formulating the critical
diffraction model based on a maximum curvature and its comparison with
experiment and simulations.

##  II NUMERICS AND EXPERIMENTS

Our recent detonation diffraction experiments and simulations in 2H2+O2+2Ar
are discussed in detail elsewhere Mevel, Xiao, and Radulescu (2017); Gallier
_et al._ (2017). The numerical simulations were performed using a realistic
chemical kinetic scheme for hydrogen combustion using AMR (Adaptive Mesh
Refinement) with a minimum grid spacing of 7 $\mu$m. This resolution of 7
$\mu$m corresponds to 40 to 90 grid points in the induction length (depending
on pressure), which is well above usual recommendations. Given that low
pressure hydrogen detonations have reaction zones longer than the induction
zones by an order of magnitude, and half-reaction zone lengths 2-3 times
larger than the induction zone lengthXiao and Radulescu (2020b), our effective
resolution is on the order of 100 to 200 grid points per half reaction length.
For reference, the recently reported state of the art simulations of Shi, Uy,
and Wen (2020) used only 24 grid points per half reaction length, a limitation
imposed by the lack of AMR. We have tested the convergence of the calculations
in straight channels for three different resolutions (7, 14 and 28 $\mu$m) in
wide channels permitting up to 10 cells and no effects on cell size were
observed. The effect of grid resolution on the diffraction behavior was not
performed due to limitation in computational resources.

Figure 1 shows three examples of supercritical, critical and subcritical
diffraction obtained in the numerics. For a fixed geometry, at a sufficiently
low pressure and long reaction zone (row a), the detonation wave is quenched
by a lateral failure wave originating at the corner, which penetrates to the
axis of symmetry. The shock and reaction zone de-couple and decay. This is the
classical failure dynamics observed in the past experiments and simulations
with realistic parameters of gas phase detonations, as also recently
documented in detail by Pintgen and Shepherd (2009); Gallier  _et al._ (2017).
At a sufficiently high pressure and short reaction zone (row c), the failure
wave cannot penetrate to the axis and the detonation wave never extinguishes
on the axis. Instead, it continues to propagate with a cellular structure. At
critical conditions intermediate to the two others (row b in the figure), re-
inititiation is observed through the amplification of lateral hotspots,
previously observed and analyzed by Jones  _et al._ (2000); Arienti and
Shepherd (2005); Shi, Uy, and Wen (2020).

Figure 1: Numerical Schlieren images adapted from Mevel, Xiao, and Radulescu
(2017) of diffracting detonations in a 2H2+O2+2Ar mixture at $T_{0}=295$K; a)
subcritical diffraction, $p_{0}=6.9$kPa, b) critical diffraction,
$p_{0}=10.3$kPa; c) supercritical diffraction $p_{0}=13.8$kPa; the height of
the computational domain is 188 mm; the time indicated below the images
corresponds to the time after the detonation exits the channel; overlaid
curves are for a weakly supported shock (green), Whitham’s inert shock model
(orange) and WSB model (blue); reproduced with permission from the authors.
Figure 2: Composite Schlieren images of detonation diffraction in 2H2+O2+2Ar
mixture at $T_{0}=295$K and $p_{0}=23$kPa, adapted from Mevel, Xiao, and
Radulescu (2017); the distance between the bottom and top walls is 200 mm;
overlaid curves are for a weakly supported shock (green), Whitham’s inert
shock model (orange) and the WSB model (blue); reproduced with permission from
the authors

The experiments showed very similar dynamics. Close to the limit, the failure
wave does not penetrate to the axis and a curved detonation wave survives
quenching (Fig. 2). At the limit (Fig. 3), re-initiation occurs from a
transverse wave amplification into a transverse detonation wave. The origin of
this surviving triple point, identifiable by the protuberance on the failing
detonation front, can be traced back to the axis in subfigure a), which also
coincides closely to the failure wave arrival at the top wall. In this sense,
this critical experiment offers a magnifying glass on the critical dynamics of
diffraction. The detonation wave survives quenching by the amplification of
transverse modes over a length scale comparable to the original channel width,
signifying that the phenomena controlling sustenance are not strictly local.
The local hotspot formation was analyzed by Jones  _et al._ (2000); Arienti
and Shepherd (2005); Shi, Uy, and Wen (2020). Of interest, Arienti (2003)
further pursued this analysis in the context of Strehlow’s ray trapping theory
Strehlow (1984). While this avenue of research appears very worthwhile in
explaining the details of the re-initiation mode and the build-up of
transverse waves, how the cellular structure impacts the detonation
diffraction phenomenon in more unstable mixtures and how potential
accompanying turbulent diffusion assists the auto-ignition phenomena Xiao and
Radulescu (2020a), this is left for future study.

Figure 3: Composite Schlieren images of detonation diffraction in 2H2+O2+2Ar
mixture at $T_{0}=295$K and $p_{0}=17$kPa, adapted from Mevel, Xiao, and
Radulescu (2017); the distance between the bottom and top walls is 200 mm;
overlaid in green is the shock shape predicted with the weakly supported shock
assumption; reproduced with permission from the authors.

Table 1 provides a summary of the limits observed experimentally and
numerically in terms of the ratio between the channel bi-thickness $W$
normalized by the ZND induction zone length. For reducing the experiments,
since the detonation prior to diffraction was found to propagate at a speed
lower than CJ due to wall losses, the channel width was normalized by the
induction zone length calculated for the conditions of the weaker lead shock.
The numerical limit of $W_{\star}/\Delta_{i}=176$ was found approximately 30%
lower than the experiments, where $W_{\star}/\Delta_{i}$ varied between 200
and 260. This variability in the experiments is due to the stochasticity,
which can be attributed to the cellular structure controlling the details of
the limiting phenomena, as illustrated in Fig. 3, for example.

Table 1: Summary of diffraction experiments and model prediction. |  $n$ in (1) |  $n$ for $\gamma=1.4$ |  $W_{\star}/\Delta_{i}$ (model) |  $W_{\star}/\Delta_{i}$ (num) |  $W_{\star}/\Delta_{i}$ (exp)
---|---|---|---|---|---
Whitham | $n_{W}=1+\frac{2}{\gamma}+\sqrt{\frac{2\gamma}{\gamma-1}}$ | 5.07 | 146 | 176 | 200-260
WSB | $n_{WBS}=3\left(\frac{\gamma+1}{\gamma}\right)$ | 5.14 | 145 | 176 | 200-260
shock support | $n_{R}=2\left(\frac{\gamma+1}{\gamma}\right)$ | 3.43 | 164 | 176 | 200-260

##  III WHITHAM’S SELF-SIMILAR GSD SOLUTION FOR THE SHOCK CURVATURE
DISTRIBUTION

Whitham’s geometric shock dynamics (GSD) provides a simple framework to
predict the dynamics of shocks affected by changes in the shock inclination
angles, such as diffraction, as well as determine the curvature evolution
Whitham (1974). For generic local shock evolution equations of the form

$\frac{S_{w}^{2}\kappa}{\dot{S}_{w}}\equiv\frac{\mathrm{d}\ln
A/\mathrm{d}x}{\mathrm{d}\ln S_{w}/\mathrm{d}x}=-n$ (1)

(i.e., $S_{w}\propto A^{-1/n}$) where $S_{w}$ is the normal speed of the shock
with respect to a medium at rest, $\kappa=\frac{\mathrm{d}\ln A}{\mathrm{d}x}$
its local curvature, $A$ a measure of the surface area of a shock element and
$n$ an arbitrary exponent, the shape of a diffracting shock over a sharp
corner is given by equations (8.95) in Whitham Whitham (1974):

$\displaystyle\frac{X}{S_{w}t}=\sqrt{\frac{n+1}{n}}\exp\left(\frac{\theta}{\sqrt{n}}\right)\sin\left(\eta-\theta\right)$
(2)
$\displaystyle\frac{Y}{S_{w}t}=\sqrt{\frac{n+1}{n}}\exp\left(\frac{\theta}{\sqrt{n}}\right)\cos\left(\eta-\theta\right)$
(3)

where $\eta$ is given by $\tan\eta=\sqrt{n}$ and $\theta$ is the angle of the
unit normal to the shock surface with the $x$-axis, see Fig. 4. These
expressions derive from a purely geometric theory for how surfaces given by a
law like (1) evolve in space. The physics are reflected by the exponent $n$,
which we treat below.

Figure 4: The self-similar solution predicted by GSD for $\gamma=1.4$: weakly
supported shock (green), Whitham’s inert shock model (orange) and WSB model
(blue).

Since the shock surface is parameterized by $\theta$ in the form
$\Lambda\left(X(\theta,t),Y(\theta,t)\right)$, its local curvature at a given
time $t$ is given by

$\kappa=\frac{X_{\theta}Y_{\theta\theta}-Y_{\theta}X_{\theta\theta}}{\left(X_{\theta}^{2}+Y_{\theta}^{2}\right)^{\frac{3}{2}}}$
(4)

yielding

$\kappa=\frac{1}{S_{w}t}\left(\frac{n}{n+1}\right)\exp\left(-\frac{\theta}{\sqrt{n}}\right)$
(5)

The curvature thus decays with time and with increasing $\theta$. The maximum
curvature occurs for $\theta=0$:

$\kappa_{0}=\frac{1}{S_{w}t}\left(\frac{n}{n+1}\right)$ (6)

Adapting these results for our diffraction problem (see Fig. 5), where the
half width of the channel is $W/2$, the time required for a transverse signal
along the shock to reach $y=W/2$ is $\frac{1}{S_{w}}\frac{W}{2}\sqrt{n}$. At
this time, the shock curvature given by (6) becomes

$\kappa_{0}=\frac{2}{W}\left(\frac{\sqrt{n}}{n+1}\right)$ (7)

Figure 5: The GSD construction for approximating the shock shape when the
corner signals meet.

##  IV CRITERION FOR DETONATION SUSTENANCE

For the detonation to survive quenching, the criterion we propose, compatible
with our experiments and simulations, is that the maximum curvature of the
wave given by (7) not exceed the critical curvature permitting steady curved
detonations. The analytical work of He and ClavinHe and Clavin (1994), Yao and
StewartYao and Stewart (1995, 1996) and Klein et al. Klein, Krok, and Shepherd
(1995) permits to express this critical curvature in closed form. We use the
expression of critical curvature obtained by He and Clavin for quasi-steady
square wave detonations with an induction time characterized by an exponential
sensitivity on temperature, which was found in very good agreement with
realistic chemistry calculations Radulescu and Lee (2002):

$\kappa_{\star}^{-1}=\Delta_{i}\frac{8{\rm
e}}{1-\gamma^{-2}}\left(\frac{E_{a}}{RT_{N}}\right)$ (8)

where $\Delta_{i}$ is the induction zone length of the CJ detonation, $T_{N}$
is the temperature behind the shock of the CJ detonation and
$\frac{E_{a}}{RT_{N}}$ is the non-dimensional activation energy characterizing
the sensitivity of the induction time to temperature. From (7), we obtain the
critical channel height for successful detonation diffraction:

$\frac{W_{\star}}{\Delta_{i}}=\frac{2\sqrt{n}}{n+1}\frac{8{\rm
e}}{1-\gamma^{-2}}\left(\frac{E_{a}}{RT_{N}}\right)$ (9)

The exponent $n$ depends on the model of shock evolution adopted, as we will
see next.

##  V THE EXPONENT $n$ FROM SHOCK EVOLUTION EQUATIONS

An exact solution for decaying shock waves yielding the exponent $n$ is not
currently available. Physically based approximations have been proposed, and
we briefly review those relevant for the present problem of diffraction.
Whitham’s model for the shock evolution equation is obtained by projecting the
shock state changes along the trajectory of a $C+$ characteristic. This model
applies to shock waves for which the rear boundary conditions play a
negligible influence on the shock dynamics. For strong shocks, Whitham Whitham
(1974) obtains the exponent $n$ given by:

$n=n_{W}=1+\frac{2}{\gamma}+\sqrt{\frac{2\gamma}{\gamma-1}}$ (10)

For $\gamma=1.4$, corresponding to the post shock state in the experiments and
numerics discussed above, this exponent is 5.07. The predictions of the
decoupled detonation are shown in the Figs. 1, 2 and 3 as the orange curves.
The model is in fair agreement with the experiments and numerics. Note that
the Whitham exponent is sometimes erroneously quoted with $+$ in the square
root instead of the $-$. This unfortunate typographical error in Bartlma and
Schroder Bartlma and Schroder (1986) has persisted in the more modern
literature Yuan  _et al._ (2020). Using the incorrect exponent leads to better
agreement with simulations and experiments, but a purely fortuitous one. The
characteristic rule has also been applied to underdriven detonations relevant
to diffraction problems by Wescott et al. Wescott, Stewart, and Bdzil (2004).
Their model assumes the detonation in quasi-steady state, and requires an
embedded sonic surface. The resulting exponent $n$ of their model for sonic
under-driven detonations is:

$n=n_{WBS}=3\left(\frac{\gamma+1}{\gamma}\right)$ (11)

The model reproduces one of the limits obtained by more rigorous perturbation
methods by Yao and StewartYao and Stewart (1996). Again, this exponent has
unfortunately also been reported erroneously (as the inverse of this
expression above) in the paper of Wescott et al. For $\gamma=1.4$,
corresponding to the post shock state in the experiments and numerics
discussed above, the correct exponent is 5.14, i.e., almost identical to the
inert shock model of Whitham. The shock prediction using this model is shown
in blue; it provides the same fair agreement as the Whitham model for this
value of $\gamma$, although they model fundamentally different phenomena. The
characteristic rule adopted in the previous two models is aimed to model the
dynamics of shock waves not influenced by rear boundary conditions. This is
physically inconsistent with the dynamics observed in experiments. For the
detonation diffraction problem, the arrival of the failure wave quenches the
chemical reactions by sudden flow expansion. The shock is subsequently
partially supported by the motion of the products, which act as a piston and
influence the shock dynamics, at least in the region immediately adjacent to
the arrival of the failure wave and conducive to reaction quenching. This
situation was analyzed for pulsed sources by Chekmarev Chekmarev (1975) and
Radulescu and Law Radulescu and Law (2007) in the context of jets issuing from
finite sized sources. These authors have found two asymptotic behaviors, the
near field dynamics of the shock were controlled by the source outflow speed,
while the far field was controlled by the free dynamics of the mass layer
bounded by inner and outer facing shocks. The early dynamics of the failed
detonation fronts can be argued to correspond to the former class. In the
detonation problem, a rear facing shock is not formed, while for the
diffraction of a purely inert shock, the rear facing shock is formed to match
the post shock state to the supersonic expansion. Indeed, the flow in the lab
frame following a detonation is subsonic, whereas it is supersonic for a
strong inert shock. Based on these physical considerations, a physical model
for shock dynamics in supported shocks is to assume that there is no time
variation in the piston support of the products, mimicking a constant rear
support. Radulescu has recently derived an exact expression linking
$\frac{\partial u}{\partial t}$ immediately behind the shock with the local
dynamics of curved shocks using Fickett and Davis’ shock change equations
Radulescu (2020). For strong inert shocks, this local balance can be written
as

$\frac{(\gamma+1)^{2}}{4\gamma}\frac{1}{\dot{S}_{w}}\frac{\partial u}{\partial
t}=2\left(\frac{\gamma+1}{\gamma}\right)+\frac{S_{w}^{2}\kappa}{\dot{S}_{w}}$
(12)

To model the weak support provided by the detonation products upon arrival of
the failure wave, we assume the evolution of the rear support to vary slower
than the shock wave response to area changes, i.e., we neglect the term on the
left-hand-side of this equation. This resulting local quasi-steady
approximation for the motion of the piston leads to a simple evolution
equation for the shock of the desired form (1), with $n$ given by

$n=n_{R}=2\left(\frac{\gamma+1}{\gamma}\right)$ (13)

For $\gamma=1.4$, corresponding to the post shock state in the experiments and
numerics discussed above, this exponent is 3.43. The shock dynamics predicted
by this truncation for the shock dynamics are shown in green in Figs. 1, 2 and
3. The simple model is found in very good agreement with both the simulations
and experiment. It can thus serve to evaluate the lateral strain rate behind
the shock when the failure wave reaches the axis, as discussed above. We note
that the truncation suggested was not derived through a rational derivation
using a systematic multi-scale approach. We note that Whitham’s model and the
WSB model, and more generally all truncations of this type, are physically
justified but do not derive systematically from first principles. Their good
performance in shock and detonation diffraction applications clearly warrants
a future effort to derive these approximations from first principles. For its
further numerical validation, future work should also attempt to evaluate each
of the terms in (12) in a ray tube perpendicular to the mean shock locus in
order to evaluate the accuracy of the balance suggested. This would
necessitate taking ensemble averages of multiple realizations for achieving a
coarse-grained description at a hydrodynamic scale, as attempted in the past
in stationary problems by Radulescu  _et al._ (2007) and Reynaud, Virot, and
Chinnayya (2017). The non-stationarity of the present problem makes this
exercise much more difficult, and it is left for future study. To further
illustrate however how the detonation diffraction problem differs from the
problem of diffraction of an inert shock, we have computed numerically the
diffraction of a Mach $M=5$ shock for $\gamma=1.4$, the same parameters as for
the reactive case. The self-similar density field is shown in Fig. 6, along
with the predictions of shock dynamics discussed above. Clearly, for the inert
shock dynamics, the shock shape is well captured by the Whitham characteristic
rule. Note however the barrel shock system that separates the supersonic
steady expansion of the flow passing the corner and the gas layered between
the inner and outer facing shocks, whose dynamics dictate the shock evolution.
This is fundamentally different from the detonation diffraction process, where
these elements are absent and are replaced by a wall vortex. The discussion of
the shock dynamics predicted by different physics has shown that the shape of
the diffracting shock is weakly dependent on which model is used, since the
conditions immediately behind the shock affect its motion. More fundamentally,
while the diffraction process is not expected to be strictly self-similar, as
the local front is more likely well approximated by the WSB model close to the
axis where the reactions are still coupled, our model when reactions decouple
and the shock is weakly supported by the motion of the detonation products,
and the Whitham inert model further from the axis for free-decay, the close
coincidence of these exponents suggests quasi-self similarity. For engineering
purposes, taking any of them should be satisfactory. Nevertheless, a formal
multi-scale analysis is warranted in the framework of the theory of Detonation
Shock Dynamics of Bdzil and Stewart (1986) formulated for a kinetic law
compatible with the long reaction zone character of hydrogen detonations and
activated induction zone, such as the model investigated by Short and Bdzil
(2003). This is well outside the scope of the present discussion, which aims
at formulating an approximate transmission criterion.

Figure 6: The density distribution during the diffraction of a $M_{w}=5$ and
$\gamma=1.4$ inert shock at a $\pi/2$ corner; overlaid curves are for a weakly
supported shock (solid line) and Whitham’s inert shock model obtained with the
characteristic rule.

##  VI Transmission criterion

With the exponent $n$ in closed form from the different approximations
discussed above, the critical channel height for detonation transmission to
open space can be obtained in closed form by substituting the expressions for
$n$ given above in (9). For our model with quasi-constant rear support,
negligible non-steady effects and negligible transverse wave structure, in the
limit of a square wave reaction zone structure, we obtain:

$\frac{W_{\star}}{\Delta_{i}}=\frac{16\sqrt{2}{\rm
e}\sqrt{\gamma^{5}(\gamma+1)}}{(2+3\gamma)(\gamma^{2}-1)}\left(\frac{E_{a}}{RT_{N}}\right)$
(14)

Similar expressions can be obtained using the other expressions for the
constant $n$. For the 2H2+O2+2Ar mixture tested, the postshock $\gamma$ is 1.4
and the reduced activation energy is 4.4, obtained from the sensitivity of the
ignition delay to temperature changes at the Von Neuman state using Cantera
for the calculations. The resulting predictions of critical channel width
using the different values of $n$ are shown in Table 1, along with the
experimental and numerical values obtained in our previous experiments and
simulations discussed above. The weakly supported shock model predicts the
critical value obtained from the simulations with an error of less than 7%.
The predictions based on Whitham and Westcott et al. models for the shock
dynamics underestimate the critical channel width by 17%, which is also quite
remarkable. All the models underpredict the experiment, for which the limit is
approximately 30% larger than for the numerics.

##  VII CONCLUSIONS

The proposed model for critical detonation diffraction relies on the
prediction of the wave curvature effected by the failure wave originating from
the diffraction corner. While the models of Whitham and Westcott, Bdzil and
Stewart are found to predict these dynamics fairly well, we propose an
improvement on the shock dynamic prediction using a weakly supported shock
model, which is found in very good agreement with experiment and simulations.
Using these simple estimates for the maximum wave curvature attained when the
failure waves meet the axis, a simple criterion for successful transmission is
that this curvature not exceed the maximum curvature that can be sustained by
a curved detonation in quasi-steady state. The closed form limits obtained for
the critical diffraction channel height are found in very good agreement with
numerics and under-predict the experiments by approximately 40%. While the
present study suggests that the diffraction of weakly unstable detonations are
well predicted by the maximum curvature of a steady ZND detonation, future
work should address how this prediction performs in much more unstable
detonations, where non-steady effects become more prominentAustin (2003) and
the turbulent structure of the reaction zone provides a mechanism to enhance
the ignition processXiao and Radulescu (2020a). Indeed, detonation attenuation
with controlled side relief in porous walled tubes Radulescu and Lee (2002);
Radulescu (2003) and weakly divergent tubes Xiao and Radulescu (2020a) showed
that the curved ZND model predictions fail with increasing instability.

## Acknowledgements

MIR acknowledges financial support provided by the Natural Sciences and
Engineering Research Council of Canada (NSERC) through the Discovery Grant
”Predictability of detonation wave dynamics in gases: experiment and model
development”.

## AVAILABILITY OF DATA

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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