We classify up to Morita equivalence all blocks whose defect groups are Suzuki $2$-groups. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field, and in fact holds up to basic Morita equivalence. As a consequence Donovan's conjecture holds for Suzuki $2$-groups. A corollary of the proof is that Suzuki Sylow $2$-subgroups of finite groups with no nontrivial odd order normal subgroup are trivial intersection.

Keywords: Morita equivalence; finite groups; block theory; Suzuki groups; Donovan's conjecture

§ INTRODUCTION

Let $p$ be a prime and $(K,\cO,k)$ be a modular system with $k$ an algebraically closed field of characteristic $p$. Donovan's conjecture, which may be stated over $\cO$ or $k$, predicts that for a given finite $p$-group $P$, there are only finitely many Morita equivalence classes of blocks of finite groups with defect groups isomorphic to $P$. Further, we may ask for classifications of Morita equivalence classes of blocks with a given defect group. Most progress so far has been for tame blocks and for abelian defect groups, and there are relatively few other classes of nonabelian $p$-groups for which the conjecture or a classification is known, aside from those admitting only nilpotent blocks. See [17] for a recent summary, and also [15] where progress is recorded.

Following [24] a Suzuki $2$-group is a non-abelian $2$-group $P$ with more than one involution for which there is $\varphi \in \Aut(P)$ permuting the involutions in $P$ transitively. It is shown in [24] that $\Omega_1(P)=Z(P)=\Phi(P)=[P,P]$, so that $P$ has exponent $4$, and a characterisation of these groups is given, placing them in four infinite series labelled $\cA$ to $\cD$. Suzuki $2$-groups of type $\cA$ have order $|Z(P)|^2$ and the others have order $|Z(P)|^3$. The Suzuki $2$-groups include the Sylow $2$-subgroups of the Suzuki nonabelian simple groups and of $PSU_3(2^n)$, which we note feature as the only examples of nonabelian simple groups with nonabelian trivial intersection Sylow $2$-subgroups (see [35]).

Here we determine the Morita equivalence classes of blocks whose defect groups are Suzuki $2$-groups. An important part of this is the observation that blocks with such defect groups are controlled, so that the classification by An in [2] of controlled $2$-blocks of quasisimple groups may be applied.

Recall for the following that a Morita equivalence is basic if it is induced by an endopermutation source bimodule (see [32]). Our main result is as follows:

Let $G$ be a finite group and $B$ be a block of $\cO G$ with defect group $P$ a Suzuki $2$-group. Then $B$ is basic Morita equivalent to one of the following:

* a block of $P \rtimes \hat{E}$ where $E$ is an odd order subgroup of $\Aut(P)$ and $\hat{E}$ is a central extension of $E$ by $Z$ with $Z \leq [\hat{E},\hat{E}]$ acting trivally on $P$;

* the principal block of $H$ for $^2B_2(2^{2n+1}) \leq H \leq \Aut({}^2B_2(2^{2n+1}))$ for some $n \geq 1$;

* a block of maximal defect of $H$ where $Z(H) \leq [H,H]$ and $PSU_3(2^n) \leq H/Z(H) \leq \Aut(PSU_3(2^n))$ for some $n \geq 2$ with $[H/Z(H):PSU_3(2^n)]$ odd.

Further, the Morita equivalence preserves the isomorphism type of the defect group, the Frobenius category and the Külshammer-Puig class (see Section <ref>).

As an almost immediate consequence we have:

Donovan's conjecture holds for Suzuki $2$-groups. In fact, there are only finitely many of basic Morita equivalence classes of blocks with defect group a given Suzuki $2$-group.

The proof of main_theorem involves a detailed analysis of the structure of groups admitting a block whose defect groups are Suzuki $2$-groups, and this analysis, together with the results of [2], gives us that Suzuki Sylow $2$-subgroups are always trivial intersection for groups with no nontrivial normal subgroup of odd order.

The structure of the paper is as follows. In Section <ref> we recall the definition of the Suzuki $2$-groups and give some properties that will be useful later. In Section <ref> we recall controlled blocks and inertial quotients, and apply the relevant results of [2]. Section <ref> contains the proof of the main result. This mainly consists of the description of the structure of what we will call reduced blocks with Suzuki $2$-groups as their defect groups. We also give the full classification of blocks with Suzuki $2$-groups of order $64$ to illustrate our main result. In Section <ref> we apply the description of reduced blocks from Section <ref> to deduce the result on the trivial intersection of Suzuki Sylow $2$-subgroups. Finally, in Section <ref> we gather some observations on invariants of blocks whose defect groups are Suzuki $2$-groups using main_theorem and known results on irreducible characters of Suzuki $2$-groups and blocks with trivial intersection defect groups.

§ SUZUKI $2$-GROUPS

We recall here the description of the classes of Suzuki $2$-groups and give some useful properties.

Write $q=2^m$ for $m \in \NN$. Let $\theta$ be a field automorphism of $\FF_q$ and define $\FF_{\theta} = \{x \in \FF_q:\theta(x)=x\}$. Now $\theta$ is given by $\theta(x)=x^{2^l}$ for some $l$.

Following [24], the series of Suzuki $2$-groups are as follows. Note that distinct $\theta$ and $\epsilon$ may and do give isomorphic groups, and that recognising isomorphism is nontrivial.

Type $\cA$: When $m$ is not a power of $2$, there exist nontrivial automorphisms $\theta$ of $\FF_q$. Define $\cA(m,\theta)$ to consist of pairs $(\a,\b)$ where $\a, \b \in \FF_q$, with multiplication given by
We have $Z(P)=\{(0,\b):\b \in \FF_q\}$. Note that the Sylow $2$-subgroups of the Suzuki simple group $^2B_2(2^{2t+1})$ are of type $\cA(2t+1,\theta)$ with $\theta(x)=x^{2^{t+1}}$, so that $\theta$ has order $m=2t+1$.

Type $\cB$: Let $m \geq 2$, $\theta$ be any automorphism of $\FF_q$, and $\epsilon \in \FF_q$ such that there is no $\rho \in \FF_q$ with $\epsilon=\rho^{-1}+\theta(\rho)$. Define $\cB(m,\theta,\epsilon)$ to consist of triples $(\a,\b,\g)$ where $\a,\b,\g \in \FF_q$, with multiplication given by

$$(\a_1,\b_1,\g_1)(\a_2,\b_2,\g_3)=(\a_1+\a_2,\b_1+\b_2,\g_1+\g_2+\a_1\theta(\a_2)+\epsilon \a \theta(\b_2)+\b_1 \theta(\b_2)).$$
We have $Z(P)=\{(0,0,\g):\g \in \FF_q\}$. Note that the Sylow $2$-subgroups of the groups $PSU_3(2^m)$ are Suzuki groups of type $\cB$.

Type $\cC$: Let $m \geq 3$ be odd, $\theta$ be the unique automorphism satisfying $2 \theta^2=1$, and $\epsilon \in \FF_q$ such that there is no $\rho \in \FF_q$ with $\epsilon=\rho^{-1}+\rho\theta(\rho^2)$. Define $\cC(m,\theta,\epsilon)$ to consist of triples $(\a,\b,\g)$ where $\a,\b,\g \in \FF_q$, with multiplication given by

$$(\a_1,\b_1,\g_1)(\a_2,\b_2,\g_3)=(\a_1+\a_2,\b_1+\b_2,\g_1+\g_2+\a_1\theta(a_2)+\epsilon \a_1^{1/2} \theta(\b_2^2)+\b_1\b_2).$$
We have $Z(P)=\{(0,0,\g):\g \in \FF_q\}$.

Type $\cD$: Let $m \geq 5$ be divisible by $5$, $\theta$ be an automorphism of $\FF_q$ of order $5$, and $\epsilon \in \FF_q$ such that there is no $\rho \in \FF_q$ with $\epsilon=\rho^{-1}+\rho\theta^4(\rho)\theta(\rho)$. Define $\cD(m,\theta,\epsilon)$ to consist of triples $(\a,\b,\g)$ where $\a,\b,\g \in \FF_q$, with multiplication given by

$$(\a_1,\b_1,\g_1)(\a_2,\b_2,\g_3)=(\a_1+\a_2,\b_1+\b_2,\g_1+\g_2+\a_1\theta(a_2)+\epsilon \theta^3(\a_1) \theta(\b_2)+\b_1\theta^2(\b_2)).$$
We have $Z(P)=\{(0,0,\g):\g \in \FF_q\}$.

A search using the SmallGroups library [9] tells us the following:

The Suzuki $2$-groups of order $2^6$ are the Sylow $2$-subgroups of $^2B_2(8)$ and $PSU_3(4)$.

A feature of Suzuki $2$-groups that helps place restrictions on the structure of groups having a block with these are defect groups is the following restriction on the nature of their normal subgroups:

Let $P$ be a Suzuki $2$-group and $Q \lhd P$.

* Suppose $P$ is of type $\cA$.

* If $|QZ(P)/Z(P)| \geq 4$, then $Z(P) \leq Q$ and $Z(Q)=Z(P)$.

* If $|QZ(P)/Z(P)| \leq 2$, then $Q$ is abelian.

* If $|Z(P)|=8$ and $|QZ(P)/Z(P)| = 2$, then $Z(P) \leq Q$.

* Suppose $P$ has type $\cB$, $\cC$ or $\cD$. If $|QZ(P)/Z(P)| \geq 2$, then $Z(P) \leq Q$ and $Z(Q)=Z(P)$.

(a) Let $P$ be of type $\cA(m,\theta)$, and let $q=2^m$. Let $(\a,\b) \in Q$. For all $x \in \FF_q$,

$$[(\a,\b),(x,0)]=(0,\a \theta(x)+x \theta(\alpha)) \in Q.$$

Note that $\tau_\a:\FF_q \rightarrow \FF_q$ given by $\tau_\a(x)=\a \theta(x)+x \theta(\alpha)$ is a group homomorphism. By <cit.> if $\a_1 \neq \a_2$, both nonzero, then $\im(\tau_{\a_1})+\im(\tau_{\a_2}) = \FF_q$. Hence if $|QZ(P)/P| \geq 4$, then since there are elements $(\a_1,\b_1), (\a_2,\b_2) \in Q \setminus Z(P)$ with $\a_1 \neq \a_2$, we must have $Z(P)=[Q,P] \leq Q$.

If $|QZ(P)/Z(P)| = 2$, then there is $\alpha \in \FF_q^\times$ such that $Q \leq \{(\alpha,\beta):\b \in \FF_q\}$ and so $Q$ is abelian since $[(\a,\b_1),(\a,\b_2)]=(0,0)$ for all $\b_1,\b_2\in\FF_q$.

Finally suppose that $m=3$ and $|QZ(P)/Z(P)| = 2$. By small_cases there is just one possibility for $P$, a Sylow $2$ subgroup of $^2B_2(8)$. The subgroup structure of $P$ may then easily be determined, to show that there are no normal subgroups isomorphic to $C_4$ or $C_4 \times C_2$.

(b) By <cit.> $P$ is an ultraspecial $2$-group, and hence semi-extraspecial, meaning that for every maximal subgroup $N$ of $Z(P)$, the group $P/N$ is extraspecial. The result then follows by <cit.> (or more explicitly <cit.>).

Let $P$ be a Suzuki $2$-group of type $\cA$ with $|Z(P)|=q=2^m$. Write $r=m_{2'}$ Then $|\Aut(P)|_{2'}=(q-1)r$ and $\Aut(P)$ possesses a subgroup $C_{q-1} \rtimes C_r$, where a generator of $C_{q-1}$ corresponds to a Singer cycle and $C_r$ corresponds to field automorphisms of $\FF_q$. This group of odd order automorphisms of $P$ acts faithfully on $Z(P)$ and on $P/Z(P)$.

Since automorphisms of $P$ inducing the trivial automorphism on $P/\Phi(P)$ have order a power of $2$, odd order automorphisms of $P$ correspond to odd order automorphisms of $P/\Phi(P) \cong (C_2)^m$. By the definition of a Suzuki $2$-group there is an automorphism of order $q-1$ permuting the nontrivial elements of $Z(P)$ transitively (corresponding to a Singer cycle), and it is clear that field automorphisms of $\FF_q$ induce automorphisms of $P$. Now it follows from [25] that $C_{q-1} \rtimes C_r$ is the only odd order subgroup of $\Aut(P/\Phi(P)) \cong GL_m(2)$ containing $C_{q-1}$ and the result follows.

§ CONTROLLED BLOCKS AND NORMAL SUBGROUPS OF SUZUKI $2$-GROUPS

A $p$-group $P$ is called resistant if every saturated fusion system on $P$ is given $P \rtimes E$ for some $p'$-group $E$. For background on fusion systems see [6] or [12], and for more on resistant $p$-groups see [34]. As noted in <cit.>, Suzuki $2$-groups are resistant since $Z(P)$ consists of the identity element and all involutions in $P$. Further the same is true for all normal subgroups of Suzuki $2$-groups once we note that abelian $p$-groups are resistant:

Let $P$ be a Suzuki $2$-group and let $Q \lhd P$. Then $Q$ is resistant.

By subgroup_structure either $Q$ is abelian or $Z(Q)$ consists precisely of the identity element and the involutions in $Q$. In either case, it follows from <cit.> that $Q$ is resistant.

Before proceeding we recall definitions of subpairs and the inertial subgroup.

A $B$-subpair is a block $B$ of a group $G$ is a pair $(Q,b_Q)$, where $Q$ is a $p$-subgroup of $G$ and $b_Q$ is a
block of $QC_G(Q)$ with Brauer correspondent $(b_Q)^G=B$. The
$B$-subpairs with $|Q|$ maximized are called the Sylow
$B$-subpairs, and they are the $B$-subpairs for which $Q$ is a defect group
for $B$. Letting $P$ be a defect group for $B$, we denote by $N_G(P,b_P)$ the stabilizer in $N_G(P)$ of $(P,b_P)$ under conjugation.

The inertial quotient of $B$ is $E=N_G(P,b_P)/PC_G(P)$, together with the action of $E$ on $P$, and is determined by the fusion system $\mathcal{F}=\mathcal{F}_{(P,b_P)}(G,B)$ for $B$, sometimes called the Frobenius category. We refer to <cit.> for background on this. Basic Morita equivalence of blocks of finite groups preserves the Frobenius category (see <cit.>).

Following the presentation in <cit.>, a Külshammer-Puig class is an element of $H^2(\Aut_\mathcal{F}(P))$, which is isomorphic to $H^2(E,k^\times)$ (see <cit.>).

By <cit.> a block with normal defect group is determined up to basic Morita equivalence by the inertial quotient and Külshammer-Puig class. When $E$ has cyclic Sylow $l$-subgroups for all primes $l$ we have that $H^2(E,k^\times)$ is trivial.

The block $B$ is controlled if the fusion system $\mathcal{F}_{(P,b_P)}(G,B)$ is the same as that of its Brauer correspondent in $N_G(P)$. Every block with resistant defect group is controlled, so we have:

Let $G$ be a finite group and $B$ be a block of $\cO G$ with defect group $P$ which is either a Suzuki $2$-group or a normal subgroup of a Suzuki $2$-group. Then $B$ is a controlled block.

Recall that $B$ is nilpotent if $\mathcal{F}_{(P,b_P)}(G,B)=\mathcal{F}_P(P)$. A controlled block is nilpotent precisely when the inertial quotient is trivial, a fact we will use frequently and without reference later on.

A block with normal defect group whose inertial quotient has cyclic Sylow $l$-subgroups for all primes $l$ is nilpotent if and only if the number $l(B)$ of simple $B \otimes_\cO k$-modules is $1$.

Controlled $2$-blocks of quasisimple groups have been described by An in <cit.>, from which we have the following, noting that the Sylow $2$-subgroups of $PSU_3(2^n)$ and $^2B_2(2^{2n+1})$ are indeed Suzuki $2$-groups:

Let $G$ be a quasisimple group and $B$ a block of $\cO G$ with defect group $P$ which is a Suzuki $2$-group or a nonabelian normal subgroup of a Suzuki $2$-group. Then $P \in \Syl_2(G)$ and $G/Z(G)$ is $PSU_3(2^n)$ or $^2B_2(2^{2n+1})$ for some $n$.

§ REDUCTIONS AND PROOF OF MAIN_THEOREM

We show that every block whose defect groups are Suzuki $2$-groups is basic Morita equivalent to what we call a reduced block. We will then show that finite groups with such a reduced block have a very restricted structure.

The following result is used in previous reductions for results concerning Morita equivalence classes of blocks, and encapsulates the use of Fong-Reynolds reductions and the Külshammer-Puig reductions [28]. Recall that a block $B$ is quasiprimitive if every block of every normal subgroup covered by $B$ is $G$-stable. In particular $B$ covers a unique block for each normal subgroup.

Let $G$ be a finite group and $B$ a block of $\cO G$ with defect group $P$. Then there is a finite group $H$ and a block $C$ of $\cO H$ such that $B$ is basic Morita equivalent to $C$, a defect group $P_H$ of $C$ is isomorphic to $P$ and:

(R1) $C$ is quasiprimitive;

(R2) If $N \lhd H$ and $C$ covers a nilpotent block of $\cO N$, then $N \leq O_p(H)Z(H)$ with $O_{p'}(N) \leq [H,H]$ cyclic. In particular $O_{p'}(G) \leq Z(G)$.

Note that $B$ and $C$ have the same Frobenius category $\mathcal{F}$, and the same Külshammer-Puig class in $H^2(\Aut_\mathcal{F}(P),k^\times)$.

We call the pair $(H,C)$, where $C$ is a block of $\cO H$, reduced if it satisfies conditions (R1) and (R2) of reduced_block. If the group is clear, then we just say $C$ is reduced.

Before proceeding we recall some definitions concerning the generalized Fitting subgroup. Details may be found in [5]. A component of $G$ is a subnormal quasisimple subgroup. The components of $G$ commute, and we define the layer $E(G)$ of $G$ to be the normal subgroup of $G$ generated by the components. The layer is a central product of the components. The Fitting subgroup $F(G)$ is the largest nilpotent normal subgroup of $G$, and is the direct product of $O_l(G)$ for all primes $l$ dividing $|G|$. The generalized Fitting subgroup is $F^*(G)=E(G)F(G)$. This has the property that $C_G(F^*(G)) \leq F^*(G)$.

Let $B$ be a quasiprimitive block of $G$ with defect group $P$. Then $B$ covers a unique block $B_E$ of $E(G)$, and this has defect group $P \cap E(G)$ (see <cit.>). Let $S$ be a component of $G$. Then since $E(G)$ is a central product of the components, $B_E$ covers a unique block $B_S$ of $S$, and this has defect group $P \cap S$.

With the notation above, if $B_S$ is nilpotent, then $B_E$ is nilpotent.

This argument may be found in the proof of <cit.>.

* The $2$-rank of the centre of a quasisimple group is at most two.

* The $2$-rank of any section of the outer automorphism group $\Out(S)$ of a nonabelian simple group $S$ is at most three. Further, if $H \leq \Out(S)$ has $2$-rank $3$, then there is $N \leq H$ with $H/N \cong C_2 \times C_2$.

This may be checked in [11], using <cit.> for a detailed description of $\Out(S)$.

Let $(G,B)$ be a reduced pair. Then one of the following:

* $P \lhd G$;

* $^2B_2(2^{2n+1}) \leq G \leq \Aut({}^2B_2(2^{2n+1}))$ for some $n \geq 1$ and $B$ is the principal block;

* $Z(G) \leq [G,G]$ and $PSU_3(2^n) \leq G/Z(G) \leq \Aut(PSU_3(2^n))$ for some $n \geq 2$, and $B$ is a block of maximal defect.

We prove main_reduction in a series of steps.

Let $(R,B)$ be reduced, and suppose that $O_2(G) \neq P$, so $O_2(G)$ is a proper subgroup of $P$.

$G$ has at least one component.

Suppose that $E(G) = 1$, so that $F^*(G)=F(G)=O_2(G)Z(G)$. Then $C_G(O_2(G))=C_G(F^*(G)) \leq F^*(G)=O_2(G)Z(G)$. In particular $Z(P) < O_2(G)$, so that $\Omega_1(O_2(G)) = Z(P)$ and $P \leq C_G(\Omega_1(O_2(G)))=:N \lhd G$.

Let $B_N$ be the (unique) block of $\cO N$ covered by $B$, and note that it has $P$ as a defect group. Write $Z=\Omega_1(O_2(G))$ and $\overline{G}=G/Z$. Note that $O_{2'}(\overline{N}) = O_{2'}(G)Z/Z$, so in particular $O_{2'}(\overline{N}) \leq Z(\overline{N})$. Let $B_{\overline{N}}$ be the unique block of $\overline{N}$ corresponding to $B_N$, so that $B_{\overline{N}}$ has defect group $\overline{P}$, an abelian group. Suppose that $E(\overline{N})=1$. Then
$F^*(\overline{N})=O_2(\overline{N})O_{2'}(\overline{N})$ and $\overline{P} \leq C_{\overline{N}}(O_2(\overline{N})) \leq O_2(\overline{N})O_{2'}(\overline{N})$. Hence $P=O_2(N)$ and $P \lhd G$, contradicting our assumption. So $E(\overline{N}) \neq 1$. Let $Y$ be a component in $E(\overline{N})$ and let $X$ be the preimage of $Y$ in $N$, so that $X$ is a central product of a component $S$ of $N$ with $Z$. Hence $S$ is a component of $G$, a contradiction.

Write $1 \neq E(G)=L_1 \cdots L_t \lhd G$, where the $L_i$ are the components of $G$. Write $B_E$ for the unique block of $E(G)$ covered by $B$, and note that this has defect group $P \cap E(G)$. Since each $L_i$ is normal in $E(G)$, we may choose a block $B_i$ of $L_i$ covered by $B_E$ with defect group $P \cap L_i$. Then no $B_i$ is nilpotent. In particular, for each $i$ we have $O_2(L_i) < P \cap L_i$.

See component_not_nilpotent. Note that blocks with central defect group are nilpotent.

$E(G) \leq T:= \langle P^g:g \in G \rangle$.

It follows from components that $E(G)=\langle (P \cap E(G))^g : g \in G \rangle \leq T$.

If $P$ is of type $\cA$ with $|Z(P)|=8$ or type $\cB$ with $|Z(P)|=4$, then there is an unique component.

Note that $P \cap E(G) \lhd P$, and that $P \cap E(G)$ is a central product of the $P \cap L_i$. Suppose $P$ is of type $\cA$. If $t \geq 2$, then $P \cap L_i$ must be cylic for some $i$, in which case $B_i$ is nilpotent, contradicting components. Hence $t=1$ in this case. Similarly if $P$ is of type $\cB$, then we have $t \leq 2$, and if $t=2$, then $P \leq L_1L_2$ with $(P \cap L_i)Z(P)/Z(P) \cong C_2 \times C_2$ for each $i$. By small_cases there is only one possibility for $P$, and it is easily seen that this cannot be written as a central product of two such groups.

If $L_i$ is a component with $P \cap L_i \leq Z(P)$, then $L_i$ is the unique component and $[P:Z(P)]=16$ (in which case $|Z(P)|=4$) or $[P:Z(P)]=8$ (in which case $|Z(P)|=8$).

Consider $L_i$ with $P \cap L_i \leq Z(P)$. Then $P \leq N_G(L_{i})$ for each $i_j$, and so $L_{i} \lhd T$.

Consider $C_T(L_i) \lhd T$, and let $C$ be a block of $C_T(L_i)$ covered by the unique block $B_T$ of $T$ covered by $B$. Note that $B_T$ has defect group $P$, and we may choose $C$ with defect group $C_P(L_i)$. Suppose that $|C_P(L_i)Z(P)/Z(P)| > 2$. Then by subgroup_structure $Z(P) \leq C_P(L_i)$, so that $P \cap L_i = Z(P) \cap L_i \leq Z(L_i)$, contradicting components. Hence $|C_P(L_i)Z(P)/Z(P)| \leq 2$. Suppose that $[P:Z(P)] \geq 2^5$. Then we have $P/C_P(L_i)Z(P) \geq 2^4$, so that $P/C_P(L_i)Z(P)$ has $2$-rank at least $4$. But $P/C_P(L_i)Z(P)$ is isomorphic to a section of $\Out(L_i/Z(L_i))$, which is impossible by rank_out_mult. We are left with $[P:Z(P)] \leq 16$. This means $P$ is either of type $\cA$ with $|Z(P)|=[P:Z(P)]=8$ or $P$ is of type $\cB$ with $[P:Z(P)]=16$ and $|Z(P)|=4$. In either case $L_i$ must be the unique component by type_AB_unique_comp.

$P \cap L_i \lhd P$ for each $i$.

By comp_defect_no_in_Z(P) we may assume $(P \cap L_i)Z(P) > Z(P)$.

Let $L_{i_1}, \ldots , L_{i_r}$ be the $G$-orbit containing $L_i$, and suppose that $r \geq 2$. Then $|(P \cap (L_{i_1} \cdots L_{i_r}))Z(P)/Z(P)|>2$ and so by subgroup_structure $Z(P) \leq L_{i_1} \cdots L_{i_r}$.

Suppose first that $Z(P) \leq Z(L_{i_1} \cdots L_{i_r})$. Since $P/Z(P)$ is abelian, it follows that $P$ normalizes each $P \cap L_{i_j}$. Suppose that $Z(P) \not\leq Z(L_{i_1} \cdots L_{i_r})$. Then each $P \cap L_{i_j}$ contains an involution in $P$ non-central in $L_{i_j}$. It follows that $P$ normalizes each $L_{i_j}$, and so normalizes a defect group $P \cap L_{i_j}$.

$G$ has only one component, i.e., $t=1$.

By comp_defect_no_in_Z(P) for each $i$ we have $(P \cap L_i)Z(P) > Z(P)$ and by components_normal $P$ normalizes each $P \cap L_i$. Suppose that $P$ is not of type $\cA$. Then by subgroup_structure $Z(P)$ is contained in a defect group for $B_i$ for each $i$. Hence if $t \geq 2$, then $Z(P) \leq Z(L_i)$ for each $i$ (since each $L_i$ contains all of the involutions in $P$). By rank_out_mult, the $2$-rank of $Z(L_i)$ is at most two, so that $|Z(P)|=4$ and $|P/Z(P)|=16$. But then $L_i$ is the unique component by type_AB_unique_comp.

Suppose that $P$ is of type $\cA$. If $t \geq 2$, then by subgroup_structure for all $i \neq j$ we have $Z(P) \leq L_iL_j$. If $t > 2$, then this means that $Z(P) \leq Z(L_i)$ for each $i$, which is impossible by rank_out_mult since $Z(P)$ has $2$-rank at least three in type $\cA$. Suppose $t=2$. If $[P \cap L_i)Z(P):Z(P)] >2$ for some $i$, then by subgroup_structure $Z(P) \leq L_i$, and as above we must have $Z(P) \leq Z(L_j)$ for each $j$, which again is impossible as $Z(P)$ has $2$-rank at least three and $Z(L_j)$ has $2$-rank at most $2$. Hence $P \cap L_i$ is abelian for each $i$ and $[P:P \cap E(G)] = [P:Z(P)]/4$. Now $G/E(G)$ is isomorphic to a subgroup of $\Out(L_1) \times \Out(L_2)$, and since $\Out(L_i)$ has $2$-rank at most $3$, this means $[P:Z(P)] \leq 2^8$. Write $[P:Z(P)]=|Z(P)|=2^m$, and note that since $P$ is of type $\cA$ we have $m \geq 3$. Since blocks with defect group $C_4 \times C_2$ are nilpotent, we must have $|(Z(P) \cap L_i)/O_2(L_i)| \geq 2^3$ (recall that by [36] a $p$-block $b$ of a finite group $X$ is nilpotent if and only if the corresponding block of $X/O_p(Z(X))$ is nilpotent). Hence $m=7$. In this case the inertial quotient of $P$ is a subgroup of $C_{127} \rtimes C_7$, which is not compatible with our configuration. Hence $t=1$ and we are done.

If $C$ is a quasiprimitive non-nilpotent block of a finite group $G_1$ with defect group $P$ of type $\cA$ with $|Z(P)|=8$, then $G_1$ cannot have a normal subgroup $H$ with $G_1/H \cong C_2 \times C_2$.

Note that we have $|\Out(P)|_{2'}=21$ and $\Out(P)$ contains a subgroup $C_7 \rtimes C_3$.

Consider the Brauer correspondent $b$ of $C$ in $N:=N_{G_1}(P)$. By <cit.> there is a unique block $\overline{b}$ of $\overline{N}:=N_{G_1}(P)/Z(P)$ dominated by $b$ (see [31] concerning domination of blocks). By <cit.> $\overline{b}$ must necessarily have defect group $P/Z(P)$. Let $(\overline{P},\overline{b}_{\overline{P}})$ be a $\overline{b}$-subpair, so that $\overline{b}$ has inertial quotient $E:=N_{\overline{N}}(\overline{P},\overline{b}_{\overline{P}})/C_{\overline{N}}(\overline{P})$. Now we cannot have $E=1$, for otherwise $\overline{b}$ would be nilpotent, in which case $l(b)=l(\overline{b})=1$ and $b$ would have trivial inertial quotient by nilpotency_criterion (noting that the inertial quotient of $b$ is a subgroup of $C_7 \rtimes C_3$, and so has trivial Schur multiplier), which would in turn imply that $B$ would have to be nilpotent since $B$ is controlled with inertial quotient a subgroup of $C_7 \rtimes C_3$. It follows from <cit.> that $\overline{P}=[\overline{P},N_{\overline{N}}(\overline{P},\overline{b}_{\overline{P}})] \times C_{\overline{P}}(N_{\overline{N}}(\overline{P},\overline{b}_{\overline{P}}))$. Suppose that $H \lhd G_1$ with $G/H_q \cong C_2 \times C_2$. Then $N=P(H \cap N)$ and $\overline{N}=\overline{P}(\overline{H} \cap \overline{N})$, and we note that $[\overline{P},N_{\overline{N}}(\overline{P},\overline{b}_{\overline{P}})] = [\overline{P},\overline{P}N_{\overline{N} \cap \overline{H}}(\overline{P},\overline{b}_{\overline{P}})] \leq \overline{N} \cap \overline{H}$. It follows that $E$ acts trivially on a subgroup of $\overline{P}$ of order $4$, and so $E$ must act trivially on $\overline{P}$, i.e., $E=1$, which we have already established cannot happen.

If $P$ is of type $\cA$ with $|Z(P)|=8$, then $P \leq {}^2B_2(2^{8}) \leq G \leq \Aut({}^2B_2(2^{8}))$.

$G$ has a unique component $L_1$, and the block $B_1$ of $L_1$ covered by $B$ has defect group $P_1:=P \cap L_1$. By type_A_auts we have $|\Out(P)|_{2'}=21$ and $\Out(P)$ contains a subgroup $C_7 \rtimes C_3$. This also acts faithfully on $Z(P)$. Since $G/L_1$ is solvable, by <cit.> $PL_1/L_1$ is a Sylow $2$-subgroup of $G/L_1$.

If $|P_1Z(P)/Z(P)| > 2$, then by subgroup_structure we have $Z(P) \leq P_1$. By Suzuki_quasisimple $L_1 \cong {}^2B_2(2^{8})$ and $P_1=P$. It follows that $O_2(G)=1$ and $F^*(G)=L_1$, so since $C_G(F^*(G)) \leq F^*(G)$ the result follows in this case.

Suppose that $P_1 \leq Z(P)$. Then $P_1 \cong C_2 \times C_2$ or $C_2 \times C_2 \times C_2$ (it cannot be $C_2$ as $B_1$ is not nilpotent). Write $\overline{P} = P/Z(P)$. We must have $F^*(G)=L_1$ and $G/L_1$ has $2$-rank three. By rank_out_mult $G/L_1$ has a normal subgroup with quotient $C_2 \times C_2$, which cannot happen by Sz(8)_even_index.

Finally, suppose that $|P_1Z(P)/Z(P)| = 2$. Now $P$ has no normal subgroup isomorphic to $C_4$ or $C_4 \times C_2$, so $P_1 \cong C_4 \times C_2 \times C_2$. In particular $P/P_1 \cong C_2 \times C_2$ and $F^*(G)=L_1Z(G)$. Since $PL_1/L_1$ is an abelian Sylow $2$-subgroup of the solvable group $G/L_1$, we have that $G$ has normal subgroups $G_1$, $G_2$ such that $|G/G_1|$ is odd and $G_2 \leq G_1$ with $G_1/G_2 \cong C_2 \times C_2$. TNow $B$ covers a unique block $C$ of $G_1$, and this also has defect group $P$. But $C$ is not nilpotent, so by Sz(8)_even_index this configuration cannot arise.

main_reduction holds.

By the previous steps, if $P \neq O_2(G)$, then we have $F^*(G) = L_1O_2(G)Z(G)$. Recall that $B_1$ is the block of $L_1$ covered by $B$ and that $P_1 : = P \cap L_1$ is a defect group for $B_1$.

Suppose that $P_1Z(P)>Z(P)$ and that $P$ is not of type $\cA$. Then by subgroup_structure $Z(P) < P_1$, so $P_1$ is a nonabelian normal subgroup of $P$, and by Suzuki_quasisimple $L_1/Z(L_1) \cong PSU_3(2^n)$ for some $m$. In this case $O_2(L_1)=1$. Since $P_i$ contains all involutions in $P$, this means that $O_2(G)=1$ and $F^*(G)=L_1$. Since $C_G(F^*(G)) \leq F^*(G)$, the result follows in this case.

Suppose that $[P_1Z(P):Z(P)]>2$ and that $P$ is of type $\cA$. Then by subgroup_structure $Z(P) \leq P_1$ with $[P:Z(P)]>2$, so $P_1$ is a nonabelian normal subgroup of $P$, and by Suzuki_quasisimple $L_1 \cong {}^2B_2(2^{2n+1})$ for some $n$. Since $P_i$ contains all involutions in $P$, this means that $O_2(G)=1$ and $F^*(G)=L_1$. Since $C_G(F^*(G)) \leq F^*(G)$, the result follows in this case.

Suppose that $[P_1Z(P):Z(P)]=2$ and that $P$ is of type $\cA$. Write $|Z(P)|=[P:Z(P)]=2^m$. In this case $P_1$ is $C_4 \times (C_2)^s$ for some $s$.

Write $H=C_G(\Omega_1(O_2(G))) \leq G$. Then $P \leq H$ and $B$ covers an unique block $B_H$ of $H$, and this has defect group $P$. We have $|O_2(G)Z(P)/Z(P)| \leq 2$, for otherwise by subgroup_structure $Z(P) \leq O_2(G)$ and $Z(P) \leq Z(H)$, which would then imply that $B_H$ is nilpotent by type_A_auts. Say $O_2(G) \cong (C_4)^i \times (C_2)^u$ for some $U,i$, where $i \in \{ 0,1 \}$.

Note that $L_1 \leq H$, that $O_2(H)=O_2(G)$, and that $O_{2'}(G)=O_{2'}(Z(G))=O_{2'}(H) \leq Z(H)$, so that $F^*(H)=F^*(G)$. Hence we have that $H/F^*(H)$ embeds in $\Out(L_1)$. By rank_out_mult $\Out(L_1)$ has $2$-rank at most three, so the $2$-rank of $P/Z(P)$ is at most five, i.e., $m \leq 5$, with equality only when $i=1$. By Sz(8) we may assume $m \neq 3$. Since $m$ is even, this leaves $m=5$. Since in this case $i=1$, we have $|P_1O_2(G)Z(P)/Z(P)| >2$. Hence by subgroup_structure we have $Z(P) \leq P_1O_2(G) \leq H$. Also note that $O_2(G)$ is not contained in $L_1$. By type_A_auts the inertial quotient of $B_H$ is a subgroup of $C_{31} \rtimes C_5$. Elements of order $31$ (given by singer cycles on $\FF_{2^5}$) act transitively on $Z(P)$ and on $P/Z(P)$, hence the inertial quotient of $B_H$ cannot contain an element of order $31$. Elements of order $5$ act as field automorphisms of $\FF_{2^5}$ on $Z(P)$ and on $P/Z(P)$. Since $5$ is prime (so that the fixed point space of a nontrivial field automorphism has order $2$), the inertial quotient cannot normalize both $O_2(G)$ and $P_1$ and so cannot contain an element of order $5$. Hence the inertial quotient of $B_H$ is trivial and $B_H$ is nilpotent. Hence we have ruled out this configuration.

Suppose that $Z(P) \leq P_1$. By comp_defect_no_in_Z(P) and Sz(8) we must have that $|Z(P)|=4$ and $[P:Z(P)]=16$, so that $P$ is of type $\cB$. But we must have $O_2(G) \leq L_1$ and $F^*(G)=L_1$, so $G/L_1$ has $2$-rank four, contradicting rank_out_mult, so this case cannot occur.

Proof of main_theorem
By reduced_block and main_reduction it suffices to consider blocks with $P \lhd G$. In this case the result follows from the main result of [26] (see also <cit.>).$\Box$

As an illustration of main_theorem, recalling small_cases, we list the Morita equivalence classes for the Suzuki groups of order $64$. Here $B_0(-)$ denotes the principal block.

Let $B$ be a block of a finite group $G$ with defect group $P$. If $P$ is a Suzuki $2$-group of order $64$, then $P$ is of type $\cA$ or $\cB$.

* If $P$ is a Suzuki group of type $\cA$, then $P$ is isomorphic to a Sylow $2$-subgroup of ${}^2B_2(8)$ and $B$ is basic Morita equivalent to one of the following:

* $\cO P$

* $\cO (P \rtimes C_7)$

* $\cO (P \rtimes C_3)$

* $\cO (P \rtimes (C_7 \rtimes C_3))$

* $B_0(\cO ({}^2B_2(8)))$

* $B_0(\cO (\Aut({}^2B_2(8))))$

* If $P$ is a Suzuki group of type $\cB$, then $P$ is isomorphic to a Sylow $2$-subgroup of $PSU_3(4)$ and $B$ is basic Morita equivalent to one of the following:

* $\cO P$

* $\cO (P \rtimes C_3)$

* $B_0(\cO PSU_3(4))$

§ TRIVIAL INTERSECTION OF SUZUKI SYLOW $2$-SUBGROUPS

A subgroup $P$ of a finite group $G$ is trivial intersection (TI) if for all $g \in G \setminus N_G(P)$ we have $P^g \cap P = 1$. Inspired by [10] we deduce the following from main_reduction.

Let $G$ be a finite group with Sylow $2$-subgroup $P$ that is a Suzuki $2$-group and suppose that $O_{2'}(G)=1$. Then $P$ is a trivial intersection in $G$.

Consider the principal block of $G$. Since the principal block of a group $H$ is nilpotent precisely when $H$ is $p$-nilpotent (that is, $[H:O_{p'}(H)]_{p}=|H|_p$), and since $O_{2'}(G)=1$, we have that $G$ satisfies (R2) of reduced_block. Condition (R1) is automatically satisfied for principal blocks, so the principal block of $G$ is reduced. The result then follows from main_reduction since all blocks listed there are TI.

§ IRREDUCIBLE CHARACTERS

Morita equivalence of $\cO$-blocks of finite groups preserves numbers of irreducible characters of each height, hence main_theorem may be used prove facts about these quantities.

Write $\Irr(B)$ for the set of irreducible characters of $G$ belonging to $B$. Writing $P$ for a defect group of $B$ and $|P|=p^d$, the height $h=h(\chi)$ of $\chi \in \Irr(B)$ is the non-negative integer $h$ such that $p^h[G:P]_p=\chi(1)_p$. Write $\Irr_h(B)$ for the set of irreducible characters in $B$ of height $h$ and $k_h(B)=|\Irr_h(B)|$.

Let $B$ be a block of a finite group $G$ with defect group $P$ that is a Suzuki $2$-group. Write $|Z(P)|=q=2^m$ and let $b$ be the Brauer correspondent block of $B$ in $N_G(P)$.

* $k_h(B)=k_h(b)$ for each $h$.

* If $P$ is not of type $\cA$, then $k_h(B)\neq 0$ precisely when $h=0$ or $h=m$.

* Suppose $P$ is of type $\cA$, defined by a field automorphism $\theta$ of $\FF_q$. Let $n$ be the order $\theta$ and $r=m/n$.

* If $n$ is odd, then $k_h(B)\neq 0$ precisely when $h=0$ or $h=(m-r)/2$.

* If $n=2$, then $k_h(B)\neq 0$ precisely when $h=0$ or $h=m/2$.

* If $n>2$ is even, then $k_h(B)\neq 0$ precisely when $h=0$, $h=(m-2r)/2$ or $h=m/2$.

By Theorem <ref> $B$ is Morita equivalent to one of the blocks listed, with the same inertial quotient and Külshammer-Puig class. The same is true of the Brauer correspondent block $b$ of $N_G(P)$. Therefore it suffices to check the result for the listed blocks. Part (a) then follows from [14].

The conclusion of (b) holds for blocks with normal defect group by [19], and (c) for blocks with normal defect group by [8] (noting that the result was first proved in [33]). Parts (b) and (c) then follow by (a).

Brauer's $k(B)$ conjecture states that $k(B) \leq |P|$. Since it is known for $p$-solvable groups, and hence for blocks with normal defect groups (see [21] for the case $p=2$), a consequence of number_irred is that the $k(B)$ conjecture holds for blocks whose defect groups are Suzuki $2$-groups. Another consequence of number_irred is that the conjecture in [18] also holds for blocks with these defect groups.

We thank Alexander Moretó for drawing our attention to [7] and [19].

[1] J. L. Alperin, Local representation theory, Cambridge University Press (1986).

[2] J. An, Controlled $2$-blocks of quasisimple groups, J. Algebra 556 (2020), 581–620.

[3] J. An and C. W. Eaton, Morita equivalence classes of blocks with extraspecial defect groups $p_+^{1+2}$, arxiv:2310.02150

[4] C. G. Ardito, Morita equivalence classes of blocks with elementary abelian defect groups of order $32$, J. Algebra 573 (2021), 297-335.

[5] M. Aschbacher, Finite group theory, Cambridge Studies in Advanced Mathematics 10, Cambridge university Press (1986).

[6] M. Aschbacher, R. Kessar and R. Oliver, Fusion systems in algebra and topolgy, London Mathematical Society Lecture Note Series 391, Cambridge University Press (2011).

[7] B. Beisiegel, Semi-extraspezielle $p$-Gruppen, Math. Z. 156 (1977), 247–254.

[8] Y. Berkovich and Z. Janko, Groups of prime power order. Vol. 2, Walter de Gruyter (2008).

[9] H. U. Besche, B. Eick, E. O'Brien and M. Horn, SmallGrp - a GAP package, Version 1.5.1 (2022), https://www.gap-system.org/Packages/smallgrp.html.

[10] H. I. Blau, On trivial intersection of cyclic Sylow subgroups, Proc. AMS 94 (1985), 572–576.

[11] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford (1985).

[12] D. A. Craven, The theory of fusion systems, Cambridge Studies in Advanced Mathematics 131, Cambridge university Press (2011).

[13] D. A. Craven and A. Glesser, Fusion systems on small $p$-groups, Trans. Amer. Math. Soc. 364 (2012), 5945–5967.

[14] C. W. Eaton, On finite groups of $p$-local rank one and conjectures of Dade and Robinson, J. Algebra 238 (2001), 623–642.

[15] C. W. Eaton, et al., Block library, https://wiki.manchester.ac.uk/blocks/

[16] C. W. Eaton, F. Eisele and M. Livesey, Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings, Math. Z. 295 (2020), 249–264.

[17] C. W. Eaton and M. Livesey, Morita equivalence classes of $2$-blocks with abelian defect groups of rank $4$, arxiv:2310.05734

[18] C. W. Eaton and A. Moretó, Extending Brauer's height zero conjecture to blocks with nonabelian defect groups, Int. Math. Res. Not. 20 (2014), 5581–5601.

[19] G. A. Fernéndez-Alcober and A. Moretó, Groups with two extreme character degrees and their normal subgroups, Trans. AMS 353 (2001), 2171–2192.

[21] D. Gluck, On the $k(GV)$-problem, J. Algebra 89 (1984), 46–55.

[22] D. Gorenstein, Finite groups, Chelsea (1980).

[23] D. Gorenstein, R. Lyons and R. Solomon, The classification of the finite simple groups, number 3, Mathematical Surveys and Monographs Vol. 40, Number 3, American Mathematical Society (1994).

[24] G. Higman, Suzuki $2$-groups, Illinois J. Math. 7 (1963), 79–-96.

[25] W. M. Kantor, Linear groups containing a singer cycle, J. Algebra,
62 (1980), 232–-234.

[26] B. Külshammer, Crossed products and blocks with normal defect groups, Comm. Alg. 13 (1985), 147–168.

[27] B. Külshammer, A remark on conjectures in modular representation theory, Arch. Math. 49 (1987), 396-399.

[28] B. Külshammer and L. Puig, Extensions of nilpotent blocks, Invent. Math. 102 (1990), 17–71.

[29] T. Le, K. Magaard and J. Moori, On the character degree of Suzuki $p$-groups of type A and C, Comm. Algebra 43 (2015), 4148–4158.

[30] M. Linckelmann, The block theory of finite group algebras, Volume II, London Math. Soc. Student Texts 92, Cambridge University Press (2018).

[31] H. Nagao and Y. Tsushima, Representations of finite groups, Academic Press (1987).

[32] L. Puig, On the local structure of Morita and Rickard Equivalences between Brauer blocks, Progr. Math., 178, Birkhäuser Verlag, Basel, 1999.

[33] I. A. Sagirov, Degrees of irreducible characters of Suzuki $2$-groups (Russian), Mat. Zemetki 66 (1999), 258–263.

[34] R. Stancu, Control of fusion in fusion systems, J. Algebra Appl. 5 (2006), 817–837.

[35] M. Suzuki, Finite groups of even order in which Sylow 2-groups are independent, Ann. of Math. 80 (1964), 58–-77.

[36] A. Watanabe, On nilpotent blocks of finite groups, J. Algebra 163 (1994), 128–134.


# Direct evaluation of attachment and detachment rate factors of atoms in
crystallizing supercooled liquids

Dinar T. Yarullin<EMAIL_ADDRESS>Bulat N. Galimzyanov
<EMAIL_ADDRESS>Anatolii V. Mokshin<EMAIL_ADDRESS>Kazan
Federal University, 420008 Kazan, Russia  Udmurt Federal Research Center of
the Ural Branch of the Russian Academy of Sciences, 426067 Izhevsk, Russia

###### Abstract

Kinetic rate factors of crystallization have a direct effect on formation and
growth of an ordered solid phase in supercooled liquids and glasses. Using
crystallizing Lennard-Jones liquid as an example, in the present work we
perform a direct quantitative estimation of values of the key crystallization
kinetic rate factors – the rate $g^{+}$ of particle attachments to a
crystalline nucleus and the rate $g^{-}$ of particle detachments from a
nucleus. We propose a numerical approach, according to which a statistical
treatment of the results of molecular dynamics simulations was performed
without using any model functions and/or fitting parameters. This approach
allows one to accurately estimate the critical nucleus size $n_{c}$. We find
that for the growing nuclei, whose sizes are larger than the critical size
$n_{c}$, the dependence of these kinetic rate factors on the nucleus size $n$
follows a power law. In the case of the subnucleation regime, when the nuclei
are smaller than $n_{c}$, the $n$-dependence of the quantity $g^{+}$ is
strongly determined by the inherent microscopic properties of a system and
this dependence cannot be described in the framework of any universal law (for
example, a power law). It has been established that the dependence of the
growth rate of a crystalline nucleus on its size goes into the stationary
regime at the sizes $n>3n_{c}$ particles.

###### keywords:

crystallization kinetics, crystal growth, nucleation, supercooled liquids,
glasses

##  1 Introduction

Crystallization and condensation are processes in which the rates of
attachment and detachment of monomers (atoms, molecules) to and from nuclei
play an important role in the nucleation and growth kinetics [1, 2]. The
details of the condensation kinetics have been well studied by experimental
and molecular dynamics simulation methods [3, 4]. Although crystallization of
supercooled liquids has also been the subject of extensive studies, the
crystallization kinetics is not well understood, especially for deep levels of
supercooling [5]. One of the main reasons for this is the absence of studies,
which are focused on accurate quantification and theoretical description of
the monomer gain and loss processes during the crystallization of supercooled
liquids and glasses [6].

The transition rate $g^{+}$ of particles from a liquid to a crystalline phase
and the detachment rate $g^{-}$ of particles from a crystalline nucleus are
the main kinetic rate factors in the theory of nucleation [1, 6]. These
kinetic factors are required to determine the rate characteristics of the
crystal nucleation and crystal growth processes [7, 8, 10, 9, 11]. Therefore,
the quantities $g^{+}$ and $g^{-}$ are included in the master equations of
well-known kinetic theories and theoretical models that describe the
nucleation and growth of crystals. The Wilson-Frenkel theory [12, 13], the
Turnbull-Fisher model [14], the Kelton-Greer model [1] and the gain-loss
theory [15] are among such the kinetic theories and models.

Direct experimental measurement of the kinetic rate factors $g^{+}$ and
$g^{-}$ for a crystallizing bulk system is a very complex task. This is due to
difficulties in tracking the trajectories of individual atoms of nano-sized
scales. The quantity $g^{+}$ can be calculated indirectly by methods based on
Turnbull or Kelton equations [16, 17, 10, 18] by using experimentally measured
self-diffusion coefficient or viscosity. Note that the accuracy of these
methods is usually insignificant, especially, for the systems at deep
supercooling. On the other hand, classical molecular dynamics simulations are
an excellent tool to extract complete information about a crystallizing system
as well as for the study of nucleation and growth processes [19, 25, 21, 24,
23, 20, 22]. In this regard, the results of molecular dynamics simulations can
be used to evaluate the kinetic rate factors.

Statistical treatment of information obtained from molecular dynamics
simulations can be performed using the mean first-passage time (MFPT) method
[26, 27, 9]. The MFPT-method is straightforward to implement in a simulation
and this method allows one to determine the activation barrier, growth curves,
and lag times [26]. As shown before, this method can be used to estimate the
value of the kinetic rate factor $g_{n_{c}}^{+}$ for the critically-sized
nucleus [Ref. [28, 29, 30]]. Another method proposed by Auer and Frenkel [31]
to compute the quantity $g_{n_{c}}^{+}$ for nuclei with the critical size
$n_{c}$ is also based on the statistical treatment of trajectories of growing
crystalline nuclei. In this method, it is assumed that the size of a nucleus
fluctuates around its critical value and this nucleus grows via the diffusive
attachment of single particles.

In the present work, we propose a simple and accurate approach for direct
evaluation of the kinetic rate factors $g^{+}(n)$ and $g^{-}(n)$. According to
this approach, the calculations can be performed as the size-dependent
quantities without using model functions and fitting parameters. We
demonstrate the efficiency of the approach for the case of study of the
crystallization kinetics of supercooled Lennard-Jones (LJ) liquid.

##  2 Approach for direct evaluation of the kinetic rate factors

Let us consider an idealized situation of mononuclear crystallization when a
single crystalline nucleus grows isotropically in a supercooled liquid. The
growth of this nucleus occurs due to the local rearrangements of parent
(disordered) phase particles, which are located near the surface of a
crystalline nucleus. Then, the quantity $k^{+}$ will determine the number of
particles (monomers) attached to the nucleus surface in a unit time. At the
same time, a crystalline nucleus can decay due to the detachment of particles
from its surface. The number of detached particles we denote as $k^{-}$. Thus,
a nucleus grows when $k^{+}>k^{-}$. On the other hand, the nucleus size
remains unchanged when $k^{+}=k^{-}$; whereas the nucleus size decreases in
the case $k^{+}<k^{-}$.

Let us suppose that the trajectories $\vec{r}_{i}(t)$ [$i=1,2,...,N$, $N$ is
the number of particles] of all particles in the system are known. Each
particle has a unique label number, which is assigned during cluster analysis
of simulation results. Then, a nucleus of the size $n(t)$ at the time $t$ can
be represented as a one-dimensional array that consists of the labels of
nucleus particles. Changes of label numbers in this array are tracked at each
simulation time step [see scheme in Fig. 1]. So, the appearance of new labels
(particles) in this array is the manifestation of the so-called gain process.
The number of such the labels will determine the value of the quantity
$k^{+}(n)$. The disappearance of labels from the array corresponds to the loss
process and the number of such the labels defines value of the quantity
$k^{-}(n)$.

Figure 1: Schematic procedure for evaluation of the quantities $k^{+}$ and
$k^{-}$ taking into account the identification (label) numbers of the
particles of a crystalline nucleus. Particles detached from the nucleus
surface are in red, whereas the attached particles are colored green.

The mononuclear crystallization scenario considered above is very specific and
it implements rarely. The polynuclear scenario is more common at the
crystallization of supercooled liquids and glasses. This scenario involves
formation of stable crystalline nuclei in a system [32]. In the case of a high
concentration of crystalline nuclei, these nuclei grow mainly due to
coalescence with each other [33]. Therefore, for evaluate the quantity $k^{+}$
we take into account only the particles, which transfer to a the crystalline
phase directly from a disordered phase. On the other hand, a crystalline
domain can decay into separate parts when incomplete coalescence of nuclei
occurs or when a nucleus has a highly ramified shape. Such the decay can lead
to a rapid increase in the number of detached particles, $k^{-}$. Therefore,
at calculation the value of the quantity $k^{-}$ we consider the particles
that transfer from the crystalline phase to the disordered phase. Exclusion
from consideration of the nuclei coalescence and nucleus decay processes
allows one to determine values of the kinetic rate factors $g^{+}$ and $g^{-}$
with high accuracy. Thus, the proposed approach is not limited to any
supercooling regime since these kinetic rate factors are calculated from their
basic definitions.

According to the basic definition [6], the kinetic rate factor $g^{+}$
characterizes the number of particles (monomers) attached to a crystalline
nucleus of size $n$ over the shortest time step $\Delta t$. The kinetic rate
factor $g^{-}$ determines the number of particles detached from $n$-sized
crystalline nucleus and transferred to the parent phase over the time step
$\Delta t$. Thus, the expressions for estimation the values of the quantities
$g^{+}$ and $g^{-}$ will have the following forms

$g^{+}(n)=\frac{\left\langle k^{+}(n)\right\rangle}{\Delta t},$ (1)
$g^{-}(n)=\frac{\left\langle k^{-}(n)\right\rangle}{\Delta t}.$ (2)

In the present work, the simulation time step is $\Delta t=0.01\,\tau$ [in the
case of argon with the parameters $m=6.63\times 10^{-26}$ kg, $\sigma=0.341$
nm, $\epsilon/k_{B}=119.8$ K and $\tau=\sigma\sqrt{m/\epsilon}$, this time
step is $\simeq 0.0215$ ps]. The brackets $\langle...\rangle$ denote averaging
over various molecular dynamics iterations. Statistical treatment of our
results is carried out over $50$ independent trajectories $n(t)$ of the
largest growing crystalline nucleus.

##  3 Results and Discussion

###  3.1 Estimation of the nucleus critical size

In the present work, we consider the crystallization of the supercooled
Lennard-Jones liquid at the temperature $T=0.5\,\epsilon/k_{B}$ and the
pressure $p=2.0\,\epsilon/\sigma^{3}$. The simulation details and applied
methods are given in the Appendix. A well-known peculiarity of one-component
LJ-system is that this system is a poor glass-former and it crystallizes
rapidly after cooling below the melting temperature $T_{m}$ [Ref. [34, 35]].
This means that it is possible to observe the crystal nucleation and crystal
growth processes in this crystallizing system at time scales available for
simulation. For the considered thermodynamic ($p$, $T$)-state, the stable
crystalline nuclei appear at the times $t>10$ $\tau$ [see Figs. 2(a) and
2(b)]. The high concentration of crystalline nuclei leads to their coalescence
at the times $t>50\,\tau$ [see Fig. 2(c)]. Fig. 2(d) shows that the system
forms a polycrystal that is the typical final structure appeared due to the
crystallization of a liquid of moderate supercooling levels [32]. To minimize
the effects related with the nuclei coalescence and nucleus decay processes on
values of the kinetic rate factors, we will consider the nuclei that grow only
within the time range $t\in[0;\,50]\,\tau$.

Figure 2: Snapshots of the system: (a) at the time $t=0$, (b) $t=10\,\tau$,
(c) $t=50\,\tau$ and (d) $t=100\,\tau$. Light blue circles indicate the
particles that form the parent (disordered) phase. The particles involved in
the formation of the crystalline phase are indicated by dark green circles.

Fig. 3(a) shows the dependence of the mean first-passage time $\bar{\tau}$ of
the largest crystalline nucleus on its size $n$ [Ref. [27, 9]]. As can be seen
from Fig. 3(a), the curve $\bar{\tau}(n)$ contains the pronounced inflection
point [28, 29, 36]. The derivative of $\bar{\tau}(n)$ over variable $n$ has
one pronounced maximum, as seen from Fig. 3(b). From the location of this
maximum on $n$-scale, we find the critical nucleus size $n_{c}\simeq(50\pm 3)$
particles. This value of the critical size is typical for spontaneously
crystallizing LJ-system at the considered ($p$, $T$)-state [10]. The average
waiting time for the critically-sized nucleus is $\tau_{c}\simeq(11.5\pm
1.2)\,\tau$ [in the case of argon with the interatomic potential parameter
$\sigma=0.341$ nm and $\epsilon/k_{B}=119.8$ K, this time $\tau_{c}$
corresponds to the value $(24.7\pm 2.58)$ ps]. Such the relatively small value
of the waiting time $\tau_{c}$ indicates on an extremely high crystal
nucleation rate in the supercooled LJ-system [34].

Figure 3: (a) Mean first-passage time $\bar{\tau}$ of the largest nucleus as a
function of the nucleus size $n$. (b) Derivative of the function
$\bar{\tau}(n)$. The arrows show the position of the inflection point in the
curve $\bar{\tau}(n)$ and the location of the maximum in the function
$\partial\bar{\tau}(n)/\partial n$.

Based on the obtained MFPT-curve shown in Fig. 3, we have calculated the
height of the nucleation barrier [26, 27]

$\beta\Delta
G^{*}=\frac{3\pi}{4}\left(\frac{n_{c}}{\tau_{c}}\right)^{2}\left[\frac{\partial\bar{\tau}(n)}{\partial
n}\Bigg{|}_{n=n_{c}}\right]^{2},$ (3)

which takes the value $\beta\Delta G^{*}\approx(2.6\pm 0.4)$ (where
$\beta=(k_{B}T)^{-1}$) for the LJ-system in the considered ($p$, $T$)-state.
In our study, the system at the high pressure $p=2.0\leavevmode\nobreak\
\epsilon/\sigma^{3}$ is considered. As we know, high pressures can accelerate
crystallization [37, 38]. As a result, we have such the small nucleation
barrier, which provides the fact that nucleation events occur in time scales
available for molecular dynamics simulation.

###  3.2 Size-dependence of the attachment rate

Fig. 4(a) shows the dependence of the reduced kinetic rate factor
$g^{+}/g^{+}_{n_{c}}$ on the reduced nucleus size $n/n_{c}$, where
$g^{+}_{n_{c}}$ is the rate of particles attachment to the critically-sized
nucleus $n_{c}$ [see Table 1]. In this dependence, two regimes can be
distinguished, which are typical for the quantity $g^{+}(n)$ [Ref. [6]]. The
first regime covers the size range $n/n_{c}\in[0;1]$ corresponding to
subcritically-sized crystalline nuclei. The second regime at the sizes
$n/n_{c}>1$ corresponds to growing supercritical nuclei.

Figure 4: (a) Reduced kinetic rate factor $g^{+}/g^{+}_{n_{c}}$ as a function of the reduced nucleus size $n/n_{c}$ for crystallizing LJ-system at the supercooling $\Delta T/T_{m}=0.43$. Our results are compared with simulation data obtained for crystallizing LJ-system at the supercooling $\Delta T/T_{m}=0.49$ [Ref. [29]] as well as for crystallizing pure Ni and binary NiAl alloy at the supercooling $\Delta T/T_{m}=0.3$ [Ref. [30]]. The wide scatter in values of the kinetic rate factor $g^{+}$ at the size $n\approx n_{c}$ is due to this quantity is evaluated near the maximum of the nucleation barrier at which the nucleus is unstable. (b) $\log[g^{+}/g^{+}_{n_{c}}]$ vs. $\log[n/n_{c}]$ curves for the considered systems. The dashed curve is obtained by Eq. (6) at $\xi=1$. Table 1: Parameters of the considered systems: supercooling $\Delta T/T_{m}$, melting temperature $T_{m}$, critical size $n_{c}$, particles attachment rate $g^{+}_{n_{c}}$ to critically-sized nucleus. The attachment rate $g_{n_{c}}^{+}$ calculated for Ni and NiAl is presented in reduced units using the Lennard-Jones parameters from Refs. [39, 40]. System | $\Delta T/T_{m}$ | $T_{m}$ | $n_{c}$ |  | $g_{n_{c}}^{+}$
---|---|---|---|---|---
LJ (our work) | $0.43$ | $0.88\,\epsilon/k_{B}$ | $50$ |  | $488\,\tau^{-1}$
LJ (from Ref. [29]) | $0.49$ | $1.15\,\epsilon/k_{B}$ | $71$ |  | $43\,\tau^{-1}$
Ni (from Ref. [30]) | $0.3$ |  $1728$ K | $52$ |  | ($1.07\times 10^{14}$) s-1 [$\simeq 21.4\,\tau^{-1}$]
NiAl (from Ref. [30]) | $0.3$ |  $1821$ K | $26$ |  | ($1.26\times 10^{12}$) s-1 [$\simeq 0.32\,\tau^{-1}$]

The found ($n/n_{c}$)-dependence of the reduced kinetic rate factor
$g^{+}/g^{+}_{n_{c}}$ is compared with the simulation data obtained for LJ-
system at the supercooling $\Delta T/T_{m}=0.49$ [at the number density
$\rho=0.95\,\sigma^{-3}$] [29], as well as with the data obtained for pure Ni
and binary NiAl-alloy at the supercooling $\Delta T/T_{m}=0.3$ [Ref. [30]].
The values of the critical size $n_{c}$ and the kinetic rate factor
$g_{n_{c}}^{+}$ estimated for the considered systems are shown in Table 1. As
seen from Fig. 4(a), all the dependencies are similar and have a common trend:
the larger the nucleus size $n$, the greater the attachment rate $g^{+}$ of
particles to the nucleus surface. Here, the values of the kinetic rate factor
$g^{+}$ estimated by our approach are obtained for nuclei with sizes up to
$10n_{c}$ particles. As far as we know, the quantity $g^{+}$ has not been
previously evaluated for nuclei with sizes more than $2n_{c}$.

In addition, we have computed the attachment rate by the mean first-passage
time $\tau(n)$ using Wedekind and Reguera method [28]:

$g^{+}(n)=B(n)/\frac{\partial\bar{\tau}(n)}{\partial n},$ (4)

where

$B(n)=-\frac{1}{P(n)}\left[\int_{n}^{2n_{c}}P(n^{\prime})dn^{\prime}-\frac{2\tau_{c}-\bar{\tau}(n)}{2\tau_{c}}\right].$
(5)

Here $P(n)$ is the probability of the formation of the largest nucleus with
size $n$ in the system. As seen from Fig. 4, the approach for direct
estimation and Eq. (4) have a similar tendency to increase $g^{+}$ with
increasing nucleus size. The agreement between the two approaches is good
despite some noise from Eq. (4). The $n$-dependence of the quantity $g^{+}$
found by Eq. (4) is also in good agreement with the data of Lundrigam’s et al.
obtained for crystallizing LJ-system with the supercooling $\Delta
T/T_{m}=0.49$ [Ref. [29]].

The ($n/n_{c}$)-dependencies of the kinetic rate factor $g^{+}$ shown on Fig.
4(a) can be well fitted by the power-law [9]

$g^{+}(n)=g_{n_{c}}^{+}\left(\frac{n}{n_{c}}\right)^{(3-\xi)/3},\,\,0<\xi\leq
3.$ (6)

Here, the exponent $\xi$ characterizes the growth regime of crystalline
nuclei. If the exponent $\xi=3$, then the quantity $g^{+}$ is independent of
the nucleus size, whereas at the value $\xi=0$ this kinetic rate factor
changes with increasing nucleus size according to $g^{+}(n)\sim n$. For LJ-
system at the considered ($p$, $T$)-state, the value of the exponent is
$\xi\simeq 1.0$. This value indicates that the attachment rate is proportional
to the nucleus surface [41]. This corresponds to the so-called ballistic model
with $g^{+}(n)\sim n^{2/3}$ in Eq. (6) [9]. In particular, this scenario is
most often realized in the case of fluid droplet growth in a supersaturated
vapor [41, 4]. As we found, the power-law (6) is valid only for nuclei with
sizes $n_{c}<n\leq 5n_{c}$ that is seen from Fig. 4(a). From the results of
cluster analysis, it follows that the nuclei with the sizes $n>5n_{c}$
interact with other small nuclei. The largest nucleus often grows through the
mechanism of restructuration and absorption of a small nucleus. In particular,
such a mechanism was observed during the crystallization of single-component
metal melts at low and deep levels of supercooling [33]. Due to the small
difference in the structure and free energy of the absorbing and absorbed
nuclei, the attachment of particles of the absorbed nucleus occurs faster than
the transition of particles from the parent phase. Note that this coalescence
mechanism is not excluded from consideration in the proposed approach for the
evaluation of kinetic rate factors. Thus, the deviation of the size dependence
of the attachment rate from the power-law $\sim n^{2/3}$ for the nuclei with
the sizes $n>5n_{c}$ is mainly due to that the largest nucleus interacts with
small nuclei. Fig. 4(b) shows that at the sizes $n<n_{c}$, the dependence
between $\log[g^{+}/g_{n_{c}}^{+}]$ and $\log[n/n_{c}]$ deviates from the
straight line with the slope $\xi\simeq 1.0$. The $n$-dependence of the
quantity $g^{+}$ calculated for subcritical sized nuclei strongly depends on
the type of the system, and this dependence is not reproducible by a power law
of form (6).

###  3.3 Competition between particle attachment and detachment processes

Fig. 5(a) shows the ($n/n_{c}$)-dependences of the kinetic rate factors
$g^{+}$ and $g^{-}$ in log-log scale.

Figure 5: (a) Particles attachment rate $g^{+}$ and particles detachment rate
$g^{-}$ as functions of the reduced nucleus size $n/n_{c}$. (b) Dependence of
the growth rate $\vartheta$ on the reduced size $n/n_{c}$ calculated by Eq.
(7). The dashed and solid curves are the theoretical results obtained from Eq.
(8) and Eq. (9), respectively. Inset: size-dependence of the free energy
$\beta\Delta G(n)$ obtained by Eq. (10).

As can be seen, the crossover of these dependences occurs in the neighborhood
of the critical size $n_{c}\simeq 50$ particles. Presence of such the
crossover point is in agreement with the general ideas of the classical
nucleation theory [6]. Obviously, for growing crystalline nuclei, the value of
the kinetic rate factor $g^{+}$ will prevail over the value of the quantity
$g^{-}$. Then, the difference between $g^{+}$ and $g^{-}$ will determine the
growth rate of nuclei in terms of the number of particles [43]

$\vartheta(n)=g^{+}(n)-g^{-}(n).$ (7)

The $n$-dependent growth rate $\vartheta$ is negative for subcritically-sized
nuclei, whereas the growth rate $\vartheta$ takes positive values for nuclei
with the critical and supercritical sizes [see Fig. 5(b)]. The found
$n$-dependence of the quantity $\vartheta$ reaches saturation at the sizes
$n>3n_{c}$. The growth of nuclei with the sizes $n>3n_{c}$ occurs due to
predominance of the kinetic rate factor $g^{+}$ over the quantity $g^{-}$.
This finding is in agreement with the data obtained before for crystallizing
Dzugutov system and binary LJ-system, where the transition to stationary
growth regime occurs at the nucleus size $n\simeq[1.7;3.0]n_{c}$ [Ref. [9]].
As a rule, such the stable growth ceases either after transition of the
crystallization process to the coalescence regime or when the system is fully
crystallized [32, 33].

The found ($n/n_{c}$)-dependence of the growth rate $\vartheta$ is reproduced
by equation [9]

$\vartheta(n)=g^{+}_{n_{c}}\left(\frac{n}{n_{c}}\right)^{2/3}\left\\{1-\exp\left(-\beta\Delta\mu\left[\left(\frac{n}{n_{c}}\right)^{1/3}-1\right]\right)\right\\}.$
(8)

In Eq. (8), the quantity $\Delta\mu$ is the difference between the chemical
potentials of the disordered and crystalline phases. As seen from Fig. 5(b),
the theoretical curve obtained from Eq. (8) reproduces only the subcritical
nuclei appearance regime, when we use $\beta\Delta\mu\approx(0.85\pm 0.1)$.
The estimated value of the difference between the chemical potentials is
$\Delta\mu\approx 0.43\leavevmode\nobreak\ \epsilon$ for LJ-system at the
considered ($p$,$T$)-state. For comparison, this value is close to the value
$\Delta\mu\approx 0.36\leavevmode\nobreak\ \epsilon$ calculated before for the
supercooled LJ-system under close thermodynamic conditions [36, 42]. In the
saturation regime, the ($n/n_{c}$)-dependence of the growth rate $\vartheta$
is reproduced by the exponential function

$\vartheta(n)=\vartheta_{st}\left\\{1-\exp\left(-a\left[\frac{n}{n_{c}}\right]\right)\right\\}.$
(9)

The growth model (9) was proposed earlier by Mydlarz and Jones to describe the
crystal growth in a stable growth regime (see Ref. [43]). As follows from this
growth model, the growth rate is close to the constant value
$\vartheta_{st}\approx 32\leavevmode\nobreak\ \tau^{-1}$ for nuclei with the
sizes $n>n_{c}$. In the case of argon, this value of $\vartheta_{st}$ is
$14.8$ ps-1. It is interesting to note that the value of the empirical
parameter $a$ coincides with the value of the chemical potential,
$a=\beta\Delta\mu\approx 0.85$. This indicates that for nuclei with the sizes
$n>n_{c}$, the nature of the relationship between the growth rate $\vartheta$
and the nucleus size $n$ is completely determined by the difference of the
chemical potentials $\Delta\mu$.

Inset on Fig. 5(b) shows the size-dependence of the Gibbs free energy
$\beta\Delta G(n)$ obtained at known values of the kinetic rate factors
$g^{+}$ and $g^{-}$. We use the expression [44]

$\beta\Delta
G(n)=-\int_{1}^{n}\ln\left[\frac{g^{+}(n^{\prime})}{g^{-}(n^{\prime}+1)}\right]dn^{\prime},$
(10)

which is obtained from the detailed balance condition

$g^{-}(n+1)N^{eq}(n+1)=g^{+}(n)N^{eq}(n)$ (11)

and Gibbs distribution for crystalline nucleus sizes

$N^{eq}(n)=N_{0}\exp[-\beta\Delta G(n)].$ (12)

Here $\Delta G(n)$ is the work required to form the $n$-sized nucleus; $N_{0}$
is the pre-exponential constant; $N^{eq}$ is the equilibrium size distribution
for the largest nucleus. The position of the maximum in the $n$-dependence of
the quantity $\beta\Delta G$ corresponds to the critically-sized nucleus
containing $52$ particles. This value almost coincides with the critical size
calculated through the derivative of the MFPT-curve (see Table 1). Moreover,
Eq. (10) yields the nucleation barrier $\beta\Delta G^{*}\approx(4.85\pm
0.35)$ that is comparable with the value $\beta\Delta G^{*}\approx(2.6\pm
0.4)$ found also through the MFPT analysis.

##  4 Concluding Remarks

Thus, direct evaluation of the kinetic rate factors $g^{+}$ and $g^{-}$ was
performed for crystalline nuclei that grow in supercooled Lennard-Jones
liquid. The size-dependences of these kinetic rate factors were determined.
Calculations were performed for nuclei with sizes up to $10n_{c}$ without
using model functions and fitting parameters. As far as we know, such the
calculations have not been performed before. The scaled kinetic rate factors
were applied to compare our results with other known simulation data obtained
for supercooled LJ-liquid as well as for supercooled Ni and NiAl melts. We
found that the ($n/n_{c}$)-dependence of the reduced kinetic rate factor
$g^{+}/g^{+}_{n_{c}}$ follows a power law at the nucleus sizes $n\geq n_{c}$.
It has been found that in the case of subcritically-sized nuclei (for
$n<n_{c}$) this power law is not observed. This finding indicates on a mixing
different nucleus growth regimes. Moreover, the dependence of the nucleus
growth rate $\vartheta$ on the nucleus size $n$ was calculated. In this
dependence, the transition to the stationary growth regime occurs for the
nuclei with the sizes $n>3n_{c}$ particles. These results are in good
agreement with theoretical calculations.

In conclusion, we note that the results of the present work can be applied to
solve the following topical tasks: development of more accurate methods for
evaluation of the rate characteristics of structural transformations in
systems with different physical and chemical properties (ionic liquids,
molecular liquids, polymer systems, colloidal solutions) [45, 46, 47];
development of severe model and/or theory for describe the nucleus size
dependence of the kinetic rate factors $g^{+}$ and $g^{-}$; quantitative
characterization and theoretical description of the decay process of
crystalline structures in supercooled liquids and glassy materials.

## Acknowledgement

This work is supported by the Russian Science Foundation (project
19-12-00022).

## Appendix: Parameters of the considered system and applied methods

The crystallization process of the supercooled Lennard-Jones liquid is
considered, where the interaction between particles is determined by the pair
potential:

$U(r)=4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right].$
(13)

Here $r$ is the distance between particles, $\sigma$ is the effective diameter
of a particle, $\epsilon$ is the parameter that characterizes the depth of the
potential well. Units of physical quantities are expressed in the terms of the
potential parameters $\sigma$ and $\epsilon$: the temperature $T$ in the unit
$\epsilon/k_{B}$, the pressure $p$ in the unit $\epsilon/\sigma^{3}$, the
kinetic rate factors $g^{+}$ and $g^{-}$ in the unit $\tau^{-1}$, where
$\tau=\sigma\sqrt{m/\epsilon}$ is the time unit, $m$ is the particle mass,
$k_{B}$ is the Boltzmann constant. For argon with the parameters $m=6.63\times
10^{-26}$ kg, $\sigma=0.341$ nm and $\epsilon/k_{B}=119.8$ K we have the value
$\tau\simeq 2.15$ ps.

The considered system contains $N=13500$ particles located inside the
simulation cubic cell with periodic boundary conditions in all three
directions. We use the standard velocity-Verlet integrator and the time-step
$\Delta t=0.01\leavevmode\nobreak\ \tau$ for integrate Newton’s equations of
motion and calculate the trajectories of particles [48]. The simulation was
performed in the isothermal-isobaric ensemble. Pressure and temperature were
controlled via the Nose-Hoover barostat and thermostat, respectively. The
damping thermostat and barostat constants were taken to be $Q_{T}=100\Delta t$
and $Q_{p}=1000\Delta t$, respectively. These values of the quantities $Q_{T}$
and $Q_{p}$ are optimal for the system at the considered ($p$,$T$)-state. The
initial system is a crystal with the face-centered cubic lattice at the
temperature $T=0\,$K. Further, the system was heated to the temperature
$T=2.5\,\epsilon/k_{B}$ at the constant pressure $p=2.0\,\epsilon/\sigma^{3}$
and, then, the system was brought to equilibrium. To prepare a supercooled
sample, the equilibrated melt was rapidly cooled with the rate
$0.04\,\epsilon/(k_{B}\tau)$ to the temperature $T=0.5\,\epsilon/k_{B}$. This
temperature corresponds to the supercooling level $\Delta T/T_{m}=0.43$, where
$\Delta T=T_{m}-T$ and the melting temperature is $T_{m}=0.88\,\epsilon/k_{B}$
on the isobar $p=2.0\,\epsilon/\sigma^{3}$. The numerical density $\rho$ of
the system at this pressure is $\simeq 0.92\,\sigma^{-3}$ [see phase diagram
of the system in Ref. [49]]. Detection of structural changes in the system
starts immediately after receiving a supercooled liquid state.

The centers of the crystalline phase are identified using cluster analysis
based on estimation of the local orientational order parameters [50].
Particles involved in the formation of the crystalline structure are detected
by ten Wolde-Frenkel condition [37]

$0.5<\left|\sum_{m=-6}^{6}\bar{q}_{6m}(i)\bar{q}_{6m}^{*}(j)\right|\leq 1,$
(14)

where

$\bar{q}_{6m}(i)=q_{6m}(i)\bigg{/}\sqrt{\sum_{m=-6}^{6}\left|q_{6m}(i)\right|^{2}}.$
(15)

Here, the $6$-fold bond order parameter is calculated through the expression

$q_{6m}(i)=\frac{1}{n_{b}(i)}\sum_{j=1}^{n_{b}(i)}Y_{6m}(\theta_{ij},\phi_{ij}),$
(16)

where $Y_{6m}(\theta_{ij},\phi_{ij})$ are the spherical harmonics with the
polar $\theta_{ij}$ and azimuthal $\phi_{ij}$ angles, $n_{b}(i)$ is the number
of neighbors for the $i$th particle. According to condition (14), the $i$th
particle is considered as involved in a crystalline phase if this particle has
four or more crystal-like bonds with their own “neighbors”. The most probable
growth trajectory – the time dependence of the nucleus size $n(t)$ – of the
largest crystalline nucleus is determined through the statistic treatment of
the cluster analysis results. Here the $50$ independent trajectories $n(t)$
were used. The values of the crystal nucleation characteristics such as the
critical size $n_{c}$ and the waiting time $\tau_{c}$ of the critically-sized
nucleus are estimated by the method of inverted averaging of nucleus growth
trajectories [27, 9].

## References

  * [1] K. F. Kelton and A. L. Greer, J. Non-Cryst. Solids 79, 295 (1986).
  * [2] K. Yasuoka, M. Matsumoto, Chem. Phys. 109, 8451 (1998).
  * [3] P. Schaaf, B. Senger, J.-C. Voegel, R. K. Bowles, H. Reiss, J. Chem. Phys. 114, 8091 (2001).
  * [4] J. Diemand, R. Angelil, K. K. Tanaka, H. Tanaka, J. Chem. Phys. 139, 074309 (2013).
  * [5] G. C. Sosso, J. Chen, J. Cox, M. Fitzner, P. Pedevilla, A. Zen, A. Michaelides, Chem. Rev. 116, 7078 (2016).
  * [6] D. Kashchiev, Nucleation: Basic Theory with Applications (Butterworth-Heinemann, Oxford, 2000).
  * [7] V. Agrawal, B. Peters, Advances in Chemical Physics 155, 97 (2014).
  * [8] F. Mura and A. Zaccone, Phys. Rev. E 93, 042803 (2016).
  * [9] A. V. Mokshin, B. N. Galimzyanov, Phys. Chem. Chem. Phys. 19, 11340 (2017).
  * [10] V. G. Baidakov, A. O. Tipeev, J. Non-Cryst. Solids 503–504, 302 (2019).
  * [11] A. M. Rodrigues, D. R. Cassar, V. M. Fokin, E. D. Zanotto, J. Non-Cryst. Solids 479, 55 (2018).
  * [12] H. A. Wilson, Philos. Mag. 50, 238–250 (1900).
  * [13] J. Frenkel, Phys. Z. Sowjetunion 1, 498 (1932).
  * [14] D. Turnbull and J. C. Fisher, J. Chem. Phys. 17, 71 (1949).
  * [15] M. C. Weinberg, W. Howard Poisl, L. Granasy, C. R. Chimie 5, 765 (2002).
  * [16] D. Turnbull and R. L. Cormia, J. Chem. Phys. 34, 820 (1961).
  * [17] K. F. Kelton, Crystal Nucleation in Liquids and Glasses (Academic, Boston, 1991).
  * [18] A. V. Mokshin, B. N. Galimzyanov, D. T. Yarullin, JETP Lett. 110, 511 (2019)
  * [19] B. J. Alder, T. E. Wainwright, J. Chem. Phys. 27, 1208 (1957).
  * [20] R. E. Ryltsev, N. M. Chtchelkatchev, J. Cryst. Growth 531, 125374 (2020).
  * [21] E. Kirova, V. Pisarev, J. Cryst. Growth 528, 125266 (2019).
  * [22] N. A. Inogamov, V. A. Khokhlov, Y. V. Petrov, VV. Zhakhovsky, Optical and Quantum Electronics 52, 63 (2020).
  * [23] L. V. Kamaeva, R. E. Ryltsev, V. I. Lad’yanov, N. M. Chtchelkatchev, J. Mol. Liquids 299, 112207 (2020).
  * [24] P. K. Galenko, A. Salhoumi, and V. Ankudinov, IOP Conf. Series: Materials Science and Engineering 529, 012035 (2019).
  * [25] J. Orava and A. L. Greer, J. Chem. Phys. 140, 214504 (2014).
  * [26] J. Wedekind, R. Strey and D. Reguera, J. Chem. Phys. 126, 134103 (2007).
  * [27] A. V. Mokshin, B. N. Galimzyanov, J. Phys. Chem. B 116, 11959 (2012).
  * [28] J. Wedekind, D. Reguera, J. Phys. Chem. B 112, 11060 (2008).
  * [29] S. E. M. Lundrigan and I. Saika-Voivod, J. Chem. Phys. 131, 104503 (2009).
  * [30] H. Song, Y. Sun, F. Zhang, C. Z. Wang, K. M. Ho, and M. I. Mendelev, Phys. Rev. Materials 2, 023401 (2018).
  * [31] S. Auer and D. Frenkel, J. Chem. Phys. 120, 3015 (2004).
  * [32] B. N. Galimzyanov, D. T. Yarullin, and A. V. Mokshin, JETP Lett. 107, 629 (2018).
  * [33] B. N. Galimzyanov, V. I. Ladyanov, A. V. Mokshin, J. Cryst. Growth. 526, 125214 (2019).
  * [34] V. G. Baidakov, K. R. Protsenko, J. Phys. Chem. B 123, 8103 (2019).
  * [35] S. Stephan, M. Thol, J. Vrabec, and H. Hasse, J. Chem. Inf. Model. 59, 4248 (2019).
  * [36] K. G. S. H. Gunawardana, X. Song, J. Chem. Phys. 148, 204506 (2018).
  * [37] P. R. ten Wolde, M. J. Ruiz-Montero, D. Frenkel, J. Chem. Phys. 104, 9932 (1996).
  * [38] K. Koperwas, F. Affouard, J. Gerges, L.-C. Valdes, K. Adrjanowicz, and M. Paluch, Phys. Rev. B 96, 224106 (2017).
  * [39] H. Heinz, R. A. Vaia, B. L. Farmer, and R. R. Naik, J. Phys. Chem. C 112, 17281 (2008).
  * [40] M. Sanati, R. C. Albers and F. J. Pinski, J. Phys. Condens. Matter 13, 5387 (2001).
  * [41] V. Volterra, A. R. Cooper, J. Non-Cryst. Solids 74, 85 (1985).
  * [42] U. R. Pedersen, J. Chem. Phys. 139, 104102 (2013).
  * [43] J. Mydlarz and A. G. Jones, Chem. Eng. J. 53, 125 (1993).
  * [44] K. F. Kelton, A. L. Greer, Nucleation in Condensed Matter: Applications in Materials and Biology (Elsevier’s Science and Technology, Oxford, UK, 2010).
  * [45] C. Huang, S. Ruan, T. Cai, and L. Yu, J. Phys. Chem. B 121, 9463 (2017).
  * [46] R. M. Khusnutdinoff, A. V. Mokshin, J. Cryst. Growth 524, 125182 (2019).
  * [47] V. V. Brazhkin, Phys. Usp. 62, 623 (2019).
  * [48] L. Verlet, Phys. Rev. 159, 98 (1967).
  * [49] A. Travesset, J. Chem. Phys. 141, 164501 (2014).
  * [50] P. J. Steinhardt, D. R. Nelson, M. Ronchetti, Phys. Rev. B 28, 784 (1983).