Mauricio Labadie
email: mauricio.labadie@gmail.com

The stabilizing effect of growth on pattern formation

Keywords: Pattern formation, Reaction-diffusion, Laplace-Beltrami operator, Global existence of solutions, Stability Mathematical Subject Classification 92C15, 35K57, 58J99, 74G25, 70K20

come

The stabilizing effect of growth on pattern formation

Introduction

Since the seminal paper of Turing [START_REF] Turing | The chemical basis of morphogenesis[END_REF], the most frequently used framework for modeling pattern formation in biological and chemical systems are reaction-diffusion systems of the form

∂u ∂t = D∆u + F(u) , D =      D 1 D 2 . . . D M      , D k > 0 , (1)
where u = u(x, t), x is the position in a domain in R N and t ≥ 0 is time. However, as Plaza et al [?] remark, this framework does not take into account the effects of domain growth and curvature, which are crucial for the development of an organism. Therefore, it is necessary to develop reaction-diffusion models that consider these two important features.

There have been several works aiming at studying the effect of growth on pattern formation. In 1995 Kondo and Asai [START_REF] Kondo | A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus[END_REF] reproduced numerically the complex behavior of patterns on the skin of Pomacanthus, a tropical fish, by just adding growth to the classical reaction-diffusion system [START_REF] Barras | Mode transitions in a model reaction-diffusion system driven by domain growth and noise[END_REF]. Based on this evidence, Meinhardt [START_REF] Meinhardt | Growth and patterning: dynamics of the stripe formation[END_REF] emphasized that this result suggests a new way to look at the process of regulatory features in embryogenesis, not only in Pomacanthus but also in other organisms like Drosophila. In 1999 Crampin et al [START_REF] Crampin | Reaction and diffusion growing domains: Scenarios for robust pattern formation[END_REF] showed in a 1-dimensional simulation that domain growth may be a mechanism for increased robustness in pattern formation. They managed to find a critical growth rate, under which there is a sequence of mode-doubling pattern transitions, and they also showed that if the growth rate is much bigger or smaller than this critical value the mode-doubling pattern dissappears. In 2004 Plaza et al [?] derived a reactiondiffusion model for two morphogens on 1 and 2-dimensional growing domains. They used this model to perform numerical calculations in squares and cones with isotropic growth, and in the light of the simulations, they concluded that growth has a stabilising effect on pattern formation. More precisely, and we quote: "New patterns can be robustly selected due to the effect of either curvature and/or growth, which would be unstable otherwise".

In 2007 Gjiorgjieva and Jacobsen [START_REF] Gjiorgjieva | Turing patterns on growing spheres: the exponential case[END_REF] studied the effect of growth on pattern formation on a 2-dimensional sphere. They showed that the solutions under slow growth are very similar to the solutions of the model of Chaplain et al [START_REF] Chaplain | Spatio-temporal pattern formation on spherical surfaces: Numerical simulation and application to solid tumour growth[END_REF] on a fixed sphere, which implies that there is a continuity link between growing and fixed patterns. But they also found something very interesting, which is worth quoting: "In general, the range of eigenmodes which yield Turing pattern formation for a growing sphere is larger than the range for a fixed sphere, which implies that growth increases the number of possible patterns. However, the dominant eigenmode determining the pattern is smaller for growing spheres [. . . ] This shows that, although a larger class of patterns is allowed for growing spheres, a lower mode is typically selected".

All these four numerical examples reinforce the conjecture that growth should have some kind of stabilising effect on pattern formation. In this paper we affirmatively answer this question, showing that (i) under an isotropic regime growth has indeed a stabilising effect on patterns, i.e. the eigenvalues of a growing domain have real parts smaller than those on fixed domains; and (ii) growth enhances the possibility of a solution to be globally defined, i.e. blow-up on growing manifolds occur later than on fixed manifolds, and if growth is fast enough we can even avoid blow-up. We will finish with a discussion of the results, showing that the classical linear stability analysis for bifurcations apply to our case, and pointing out the possible applications of our findings to regulatory dynamics in pattern formation, embryogenesis and tumor growth.

Main results

Reaction-diffusion systems on growing manifolds

Let us define the mathematical objects we will work with throughout this paper. Definition 1 A manifold M will be for us a smooth (C ∞ ), compact, connected, oriented Riemannian manifold without boundary. We will denote its parametrisation by

X : Ω ⊂ R n -→ M ξ = (ξ 1 , . . . , ξ n ) -→ X(ξ)
and its metric by (g ij (ξ)).

The following definition is standard and well-known, but it is necessary for the sake of completeness. Moreover, it will allow us to set our notation.

Definition 2

The Laplace-Beltrami operator on a Riemannian manifold M with parametrisation (ξ, t) = (ξ 1 , . . . ξ n , t) and metric (g ij (ξ, t)) is (using the sum convention on repeated indices)

∆ M φ := 1 √ g ∂ ξ j [ √ g g ij ∂ ξ i φ] , (2)
where (g ij ) = (g ij ) -1 and g = det(g ij ).

Definition 3 A f growing manifold is a monoparametric family of manifolds (M t ) t≥0 such that for any t ≥ 0:

• M t is a manifold with metric (g ij (ξ, t)).

• The function t → g(t, ξ) = det(g ij (t, ξ)) is strictly increasing.

• The parametrisation X(ξ, t) is C ∞ in both variables ξ and t.

• The mapping t → g(ξ, t) := det(g ij (ξ, t)) is strictly increasing.

We will use the notation ∆ Mt and ∆ M to emphasize the time dependence or independence of the coefficients of the Laplace-Beltrami operator, respectively. ∞) is a growth function or growth factor if it is a C 1 function satisfying ρ(0) = 1 and ρ(t) ≥ 0 for all t ≥ 0.

Definition 4 A function ρ : [0, ∞) → [1,
Definition 5 A growing manifold (M t ) t≥0 has isotropic growth if there is a growth function ρ(t) and a manifold M such that M t := ρ(t)M, meaning that if X(ξ, t) is the parametrisation of M t then there is a parametrisation X of M such that X(ξ, t) = ρ(t) X(ξ).

It is important to keep in mind that at time t = T the growing manifold M 0≤t≤T coincides with the fixed manifold M T , but they are "dynamically" different. Indeed, in the growing manifold M 0≤t≤T the growth dynamics is included, whereas in the fixed manifold M T growth does not play any role.

Theorem 1 Let (M t ) t≥0 be a growing manifold with metric (g ij (ξ, t)). Under the hypotheses of Fick's law of diffusion and conservation of mass any reaction-diffusion system on M t has the form

∂ t u = D∆ Mt u -∂ t [log g(t, ξ) ]u + F(u) , (3)
where the Laplace-Beltrami operator ∆ Mt is given in (2). In the case of isotropic growth we have

∂ t u = D ρ 2 (t) ∆ M u -n ρ(t) ρ(t) u + F(u) , (4)
where the coefficients of ∆ M do not depend on time.

Definition 6 For a growing manifold (M t ) t≥0 we define its growth rate as

c(t, ξ) := ∂ t [log g(t, ξ) ] .
Observe that since t → g(t, ξ) is strictly increasing it follows that t → c(t, ξ) is strictly increasing as well. Theorem 1 says that in the case of isotropic growth the growth rate is independent of the spacial variable ξ and takes the form

c(t) = n ρ(t) ρ(t) .
In particular, if the growth function is exponential, i.e. ρ(t) = ae rt then c(t) := nr, hence the growth rate is constant. Note also that in the case of a 2-dimensional growing manifold (S t ) t≥0 with orthogonal tangent vectors we have

(g ij ) = h 2 1 0 0 h 2 2
and thus we recover the equations in Plaza et al [?]:

∂ t u = D∆ St u -∂ t [log(h 1 h 2 )]u + F(u) , ∂ t u = D ρ 2 (t) ∆ S u -2 ρ(t) ρ(t) u + F(u) .

Properties of solutions: existence and uniqueness

We will prove that in the case of isotropic growth, the reaction-diffusion system (4) has a timelocal unique solution. Moreover, if the initial condition is continuous and the nonlinearity

F(u) is C ∞ then the local solution is C ∞ for positive times.
Theorem 2 There is a time T > 0 such that the reaction-diffusion system (4) with initial condition u 0 ∈ C M, R M has a unique solution

u(t) ∈ C [0, T ], C M, R M . Theorem 3 If F : R M → R M is C ∞ then u(t) ∈ C ∞ M × (0, T ], R M .

The anti-blow-up effect of growth

We will show that under specific conditions, depending on the initial condition u 0 and the nonlinearity F(u), the locally defined solution u(t) is in fact globally defined. Moreover, those conditions are less restrictive on growing manifolds than on fixed manifolds, which implies that growth has an enhancing effect on the regularity of solutions.

Theorem 4 Let (M t ) t≥0 be an isotropic growing manifold with growth rate c(t). Suppose that the initial condition u 0 of the reaction-diffusion system (4) lies in C M, R M and takes its values inside the rectangle R = (-1, 1) M . Suppose further that for all (z, t)

∈ ∂R × [0, ∞) we have F(z) • n(z) < c(t) ∀t ≥ 0 , (5)
where n(z) is the outer normal at z. Then the solution u(t) of ( 4) is global and bounded, i.e. it exists for all times t ≥ 0 and takes its values inside R. In particular, there is no blow-up whenever (5) holds.

From Theorem 4, if the growth rate is sufficiently big to satisfy

c(t) > sup{ F(z) : z ∈ ∂R}
then the solution is globally bounded, which implies that there is no blow-up. Notice that since the growth rate c(t) is increasing in n, the dimension of the space enhances the regularity of solutions.

The next two examples show a direct consequence of Theorem 4 on scalar equations: if on a fixed domain we have blow-up at time t 1 then on the corresponding growing domain we have blow-up at time t 2 > t 1 . Moreover, if growth is fast enough then a solution that blows up on a fixed domain is actually globally defined on the corresponding growing domain. This illustrates the anti-blow-up effect of growth on pattern formation: on growing domains the blow-up occurs later than on fixed domains, and it could even do not occur at all.

Example 1 Consider the scalar ODE

u = u 2 , u(0) = u 0 > 0 , whose solution is u(t) = 1 u 0 -t -1
, which blows up when t → t 1 := 1/u 0 . Now consider the equivalent problem on a growing domain, i.e. suppose u(t) is a scalar, homogeneous solution of (4). Then u solves the scalar ODE

u = -c(t)u + u 2 , u(0) = u 0 > 0 . Under the change of variables v(t) := e R t 0 c(s)ds u(t) we have v = e - R t 0 c(s)ds v 2 ,
and since

c(t) = n ρ(t) ρ(t) it follows that e - R t 0 c(s)ds = 1 ρ n (t)
,

which implies that v = 1 ρ n (t) v 2 .
In consequence,

v(t) = 1 u 0 - t 0 ds ρ n (s) -1 ,
which blows up at time t 2 , where t 2 is defined as

t 2 0 dt ρ n (t) = 1 u 0 .
However, since ρ(t) > 1 for t > 0 it follows that Let us study some special cases of growth in 1D, i.e. n = 1.

t 1 0 dt ρ n (t) < t 1 := 1 u 0 . Therefore (a) t 2 > t 1 := 1/u 0 , (b)
• If there is no growth then ρ(t) = 1 and t 2 = t 1 .

• If growth is linear then ρ(t) = 1 + αt, α > 0. In consequence,

t 2 = 1 α e α/u 0 -1 = 1 u 0 + 1 α ∞ n=2 1 n! α u 0 n > 1 u 0 = t 1 .
• Suppose that we have quadratic growth. If

ρ(t) = 1 + βt 2 and u 0 > 2 √ β/π then t 2 = 1 √ β tan √ β u 0 > 1 u 0 = t 1 . However, if u 0 ≤ 2 √ β/π then ∞ 0 dt 1 + βt 2 = π 2 √ β < 1 r ≤ 1 u 0 .
Therefore t 2 = ∞, i.e. there is no blow-up. Let us consider another quadratic growth:

ρ(t) = (1 + t) 2 . If u 0 > 1 then t 2 = 1 u 0 -1 > 1 u 0 = t 1 .
However, if u 0 ≤ 1 then t 2 = ∞, i.e. there is no blow-up.

• Suppose that growth is exponential, i.e. ρ(t) = e rt , r > 0. If r < u 0 then

t 2 = - 1 r log 1 - r u 0 = 1 u 0 + 1 r ∞ n=2 1 n r u 0 n > 1 u 0 = t 1 . However, if r ≥ u 0 then ∞ 0 e -rt dt = 1 r ≤ 1 u 0 .
Therefore t 2 = ∞, i.e. there is no blow-up.

Example 2 Suppose u(t, ξ) is a local solution of the scalar reaction-diffusion equation

∂ t u = 1 ρ 2 (t) ∆ M u -c(t)u + u 2 , u(0, ξ) = u 0 (ξ) > 0 . Define η(t) := M u(t, ξ) dΩ , η(0) = M u 0 (ξ) dΩ > 0 .
On the one hand,

η(t) = M u(t, ξ) dΩ ≤ α M u 2 (t, ξ) dΩ 1/2 , α = |M|, .
On the other hand,

η = M ∂ t u(t, x) dΩ = M 1 ρ 2 (t) ∆ M u -c(t)u + u 2 dΩ = M -c(t)u + u 2 dΩ .
Therefore,

η ≥ -c(t)η + αη 2 .
Under the change of variables

ζ(t) := e R t 0 c(s)ds η(t) we obtain ζ ≥ e - R t 0 c(s)ds ζ 2 ,
which implies that

ζ(t) ≥ 1 η(0) -α t 0 ds ρ n (s) -1 .
Let t 1 be the blow-up time for the fixed manifold and t 2 the blow-up time for the growing manifold. As in Example 2, using ρ(t) > 1 for t > 0 we have (a) t 2 > t 1 = (αη(0)) -1 , (b) t 1 and t 2 are decreasing functions of the initial condition η(0), (c) t 2 is an increasing function on the spatial dimension n, and (d) if growth is sufficiently fast, i.e. if the growth function ρ(t) satisfies

∞ 0 dt ρ n (t) ≤ 1 αη(0)
then t 2 = ∞, i.e. there is no blow-up on the growing manifold. In particular, for an exponential growth ρ(t) = e rt such that nr > αη(0), i.e.

r >

|M| n M u 0 (ξ) dΩ ,

we have blow-up on the fixed manifold but not on the growing manifold.

The stabilising effect of growth

Under isotropic regimes, growth has a stabilising effect on pattern formation.

Theorem 5 Let (M t ) 0≤t≤T be an isotropic growing manifold with growth rate c(t). Define S := M T and notice that we will use the notation S for the fixed manifold and M T for the final stage of the growing manifold (M t ) 0≤t≤T . Then λ is an eigenvalue of the reaction-diffusion operator on S,

L S := D ρ 2 (T ) ∆ S + dF(0)
if and only if λ -c(T ) is an eigenvalue of the corresponding operator on M T ,

L M T := D ρ 2 (T ) ∆ M T -c(T )I + dF(0) .
Theorem 5 says that when we compare the spectra of L S and L M T on the same manifold S = M T we obtain spectrum(L M T ) = spectrum(L S ) -c(T ) .

Therefore, growth shifts the eigenvalues to the left in the complex plane, which is indeed as a stabilising effect since the real parts are smaller. Moreover, this shift is exactly the growth rate c(T ) > 0, which implies that the faster growth is, the more stable the patterns are. It is important to remark that, as far as we know, Theorem 5 is the first analytic proof of the stabilising effect of growth on pattern formation.

3 Proof of Theorem 1

We will divide the proof in three parts.

Parametrisation and Riemannian metric

Let M t be a growing manifold parametrized by X = X(ξ, t), and suppose that the motion

ψ t : Ω ⊂ R n -→ M t ξ = (ξ 1 , . . . , ξ n ) -→ ψ t (ξ) = X(ξ, t)
is C ∞ in both variables (ξ, t). Remark that the manifolds M t are supposed to be all embedded in the same Euclidean space in order to have the motion ψ t (ξ) well defined as a function of t, and that for any fixed t ≥ 0 the function ψ t (ξ) is a parametrisation for M t .

We will suppose that for all t ≥ 0 the manifold M t has a C ∞ metric (g ij (ξ, t)) with the following properties:

(a) Symmetric: g ij = g ji for all i, j.

(b) Positive definite: g ij v i v j > 0 for all v = (v 1 , . . . , v n ) = 0.

The general model with growth and curvature

Let Ω(t) be a domain in M t with boundary ∂Ω(t). Suppose that the parametrisation of Ω(t) is

Ω(t) := ψ t ( Ω) = X( Ω, t) , ∂Ω(t) := ψ t (∂ Ω) = X(∂ Ω, t) ,
where Ω is a domain in R n .

Suppose that φ = φ(X, t) denotes the concentration (given in molecules per unit area) of a morphogen (i.e. a chemical substance) at a point X ∈ M t , and let J be the flux vector of the molecules φ. The Fick's law of diffusion states that the flux vector J of the molecules is proportional to the gradient of the concentration of the molecules, i.e. J = -D∇φ , where D is the diffusion coefficient, which is assumed to be constant. The law of conservation of mass states that the rate of change on the concentration of molecules in Ω(t) is equal to the net flux of molecules on the boundary ∂Ω(t), i.e.

d dt Ω(t) φ dV = - ∂Ω(t) J, n dS . (6)
The minus sign comes from the fact that n is the unit outward normal on ∂Ω(t) and therefore J, n is the exit flux.

Using the last two relations we obtain

d dt Ω(t) φ dV = D ∂Ω(t)
∇φ| ∂Ω(t) , n dS .

For the sake of clarity let us recall each term in [START_REF] Jost | Riemannian geometry and geometric analysis[END_REF].

(a) dV is the volume element for the n-manifold Ω(t); in local coordinates we have dV = √ g dξ.

(b) dS the "area" (i.e. the (n -1)-volume) element for the (n -1)-manifold ∂Ω(t).

(c) ∇φ| ∂Ω(t) is the restriction of the vector field ∇φ to ∂Ω(t) and n is the unit outward normal.

(d) If T X Ω(t) is the tangent plane at X ∈ Ω(t) then the inner product •, • is defined as u, v := g ij (ξ, t)u i v j for any u, v in T X Ω(t). Now we calculate both sides of [START_REF] Jost | Riemannian geometry and geometric analysis[END_REF]. For the left hand side, using the change of variables φ(ξ, t) := φ(X(ξ, t), t) we obtain

d dt Ω(t) φ(X, t) dV = d dt Ω φ(ξ, t) √ g dξ = Ω[∂ t φ √ g + φ∂ t √ g ]dξ = Ω ∂ t φ + φ ∂ t √ g √ g √ g dξ = Ω(t) ∂ t φ + φ ∂ t √ g √ g dV .
For the right hand side in [START_REF] Jost | Riemannian geometry and geometric analysis[END_REF] we will use Stokes' theorem, which in the general case of on a n-manifold N with boundary ∂N can be written as

M dω = ∂M ω , ( 8
)
where

ω is a (k -1) differential form, k ≤ n. If ω is the (n -1)-form ω = F | ∂Ω(t) , n dS
with F a vector field on the submanifold Ω(t) ⊂ M t then [START_REF] Kondo | A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus[END_REF] becomes

Ω(t) div(F ) dV = ∂Ω(t) F | ∂Ω(t) , n dS . (9)
In the light of ( 9) we have that

∂Ω(t) ∇φ| ∂Ω(t) , n dS = Ω(t) div(∇φ) dV .
Putting all two pieces together in [START_REF] Jost | Riemannian geometry and geometric analysis[END_REF] we obtain

Ω(t) ∂ t φ + φ ∂ t (log √ g) -Ddiv(∇ φ) dV = 0 .
Recall that Ω(t) was an arbitrary domain in M t , and let us drop the tildes for a more convenient notation. Therefore we obtain the equation of the diffusive part of the model:

∂ t φ = D div(∇φ) -∂ t [log √ g]φ , (10)
where the operator div(∇φ) is the Laplace-Beltrami operator Mt we introduced in Definition 2. Now consider a morphogen vector u = (u 1 , . . . , u M ) and suppose that there is an extra term F(u) that models reaction kinetics, i.e. the chemical interactions between the morphogens. Then the mass balance equation ( 6) takes the form

d dt Ω(t) u dS = - ∂Ω(t) J, n dS + Ω(t) F(u) dS . ( 11
)
Now let us assume that the flux is J = -D∇u, where the matrix of diffusivities D is diagonal, i.e.

D =    D 1 . . . D M    ,
and with constant and positive coefficients D i . Under these assumptions the equation ( 11) takes the form

d dt Ω(t) u dS = D ∂Ω(t)
∇u| ∂Ω(t) , n dS +

Ω(t) F(u) dS . ( 12
)
Notice that in [START_REF] Rosenberg | The Laplacian on a Riemannian manifold[END_REF] each morphogen diffuses independently of the other and without obstacles. Therefore we can take separately the equations for each of the components u i of u in [START_REF] Rosenberg | The Laplacian on a Riemannian manifold[END_REF] and repeat the former calculations using φ = u i . Proceeding that way we obtain the general model for a reaction-diffusion system on the growing manifold M t ,

∂ t u = D∆ Mt u -∂ t [log √ g ]u + F(u) .
This proves part (a) of Theorem 1.

The isotropic growth model

In the case of isotropic growth we have

X(ξ, t) = ρ(t) X(ξ) ,
which implies the following identities:

g ij (ξ, t) = ρ 2 (t) g ij (ξ) , (13)
g ij (ξ, t) = 1 ρ 2 (t) g ij (ξ) , √ g = ρ n g , ∆ Mt = 1 ρ 2 (t) ∆ M .
If we substitute the relations [START_REF] Taylor | Partial differential equations III[END_REF] in the general model given in ( 4) we obtain the model for a n-dimensional manifold with isotropic growth:

∂ t u = D ρ 2 (t) ∆ M u -n ρ(t) ρ(t) u + F(u) .
This proves part (b) of Theorem 1 and concludes its proof.

Proof of Theorem 2

Let M be a manifold and consider the reaction-diffusion equation

∂ t u = Lu + G(t, u) , u(0) = u 0 , ( 14
)
where L is a second-order elliptic operator and the nonlinearity G is C ∞ in its arguments. Let X be a Banach space such that the following conditions hold:

1. X is a space of functions u : M → R M .

2. e tL : X → X is a strongly continuous semigroup for t ≥ 0.

3. There exists a constant C > 0 such that e tL ≤ C for all t ≥ 0.

The nonlinearity

G : X -→ X u -→ G(t, u)
is locally Lipschitz in u, uniformly in t.

Remark 1

The space X we have in mind is C M, R M , but there are other possible choices. Indeed, if we ask the nonlinearity G(t, u) to be C ∞ , bounded and with derivatives bounded then L p M, R M and H k M, R M are suitable spaces as well.

Lemma 1 The reaction-diffusion system (4) can be reduced to the system [START_REF] Turing | The chemical basis of morphogenesis[END_REF] with

L := D∆ M , G(t, u) := -nρ(t) ρ(t)u + ρ 2 (t)F(u) . Moreover, if F(z) is locally Lipschitz in z ∈ R M then G(t, z) is also locally Lipschitz in z ∈ R M , uniformly in t ∈ [0, T ].
Proof: Define the change of variables

s(t) := t 0 dr ρ 2 (r) . (15)
Then for any function f (t) we have that

∂ s f = ρ 2 (t)∂ t f .
Multiply the system (4) by ρ 2 (t) and define u(s, x) := u(t(s), x). Then the system (4) takes the equivalent form

∂ s u = D∆ M u + G(s, u) , where G(t, u) = -nρ(t) ρ(t) u + ρ 2 (t)F( u) .
Renaming the variables (s, u) as (t, u) we obtain [START_REF] Turing | The chemical basis of morphogenesis[END_REF].

Now we have all the elements to prove Theorem 2. By the Lemma 1 the problem ( 4) is equivalent to [START_REF] Turing | The chemical basis of morphogenesis[END_REF]. Moreover, ( 14) can be expressed in the integral form

u(t) = e tL u 0 + t 0 e (t-s)L G(s, u(s)) ds . (16)
Define the operator

Ψu(t) := e tL u 0 + t 0 e (t-s)L G(s, u(s)) ds on the Banach space C ([0, T ], X) with norm u(t) X := sup s∈[0,T ] u(s) ,
where • is the norm in R M . Now fix α > 0 and define

Z := {u ∈ C ([0, T ], X) : u(0) = u 0 , u(t) -u 0 X ≤ α} .
The final time T > 0 will be chosen later in order to have that Ψ :

Z → Z is a contraction. Observe that Z is a closed subset of C ([0, T ], X). Moreover, Z is bounded because if u ∈ Z then for all t ∈ [0, T ] we have u(t) X ≤ u(t) -u 0 X + u 0 X ≤ α + u 0 X .
Now we affirm that there is a constant K 1 > 0 such that if u ∈ Z then G(t, u(t)) X ≤ K 1 for all t ∈ [0, T ]. Indeed, recall that G(t, u) is locally Lipschitz continuous uniformly in t. Hence, if K is the Lipschitz constant for G then for any u ∈ Z we have

G(t, u(t)) X ≤ G(t, u(t)) -G(t, u 0 ) X + G(t, u 0 ) X ≤ K u(t) -u 0 X + G(t, u 0 ) X ≤ Kα + G(t, u 0 ) X .
The assumptions on the space X imply that there exists a C > 0 such that

e tL L(X) ≤ C .
This fact and the boundedness of G imply that

t 0 e (t-s)L G(s, u(s)) ds X ≤ K 1 Ct .
Therefore, it is possible to choose T > 0 such that K 1 CT ≤ α/2. Moreover, from the assumpions on X we have that e tL is a strongly continuous semigroup for t ≥ 0, which implies that there is a T > 0 such that e tL u 0 -u 0 X ≤ α/2 for all t ∈ [0, T ].

In conclusion, if we choose T > 0 is sufficiently small then Ψu(t) -u 0 X ≤ α, which implies that Ψ(Z) ⊂ Z.

Let us find the conditions under which Ψ is a contraction. If we calculate

Ψu(t) -Ψv(t) X = t 0 e (t-s)L G(s, u(s)) -G(s, v(s)) ds X ≤ CKt sup s∈[0,t] u(s) -v(s) X ≤ CKT u(t) -v(t) X
we can see that Ψ : Z → Z will be a contraction if we choose T > 0 such that CKT < 1. In that case we obtain that there is a unique solution u(t) ∈ C ([0, T ], X) of ( 16), or equivalently a unique solution of ( 14). This completes the proof of Theorem 2.

Proof of Theorem 3

Fix t ∈ (0, T ] and consider a solution u(t) of ( 14) written in its integral form (16). Recall two properties of the Laplace-Beltrami operator. First, the map

e t M : C(M) → C 1 (M)
is continuous for all t > 0, and second, there exists a constant C > 0 such that

e t M L(C(M),C 1 (M)) ≤ Ct -1/2
(see Taylor [?], p. 274). This implies that the operator L satisfies the same type of inequality, i.e.,

e tL L(C[M,R M ],C 1 [M,R M ]) ≤ Ct -1/2
, with a bigger constant C > 0, of course, that depends on the diffusion coefficients of the matrix D. Consequently, for any t 1 ∈ (0, t ] and u 0 ∈ C M, R M we have that u(t 1 ) ∈ C 1 M, R M . Now, if we consider u(t 1 ) as a new initial condition we have that u(t 2 ) ∈ C 2 M, R M for any t 2 ∈ (t 1 , t ]. Repeating this iterative argument we can construct a sequence

0 < t 1 < t 2 < • • • < t n → t such that u(t n ) ∈ C n M, R M for any t n ∈ (t n-1 , t ].
In the limit we get that u(t ) ∈ C ∞ M, R M for any t ∈ (0, T ].

Concerning the time derivatives, recall that if u(ξ, t) is a solution of (4) then

∂ t u = D ρ 2 (t) ∆ M u -n ρ(t) ρ(t) u + F(u) .
Therefore

(t, x) is C ∞ in time. In conclusion, u(ξ, t) ∈ C ∞ (0, T ] × M, R M .

Proof of Theorem 4

Lemma 2 Let M be a manifold and consider the reaction-diffusion system

∂ t u = D∆ M u + F(u) , u(ξ, t) = u 0 (ξ) . (17)
Suppose that u 0 (ξ) ∈ C M, R M and that it takes its values inside the rectangle

R = M j=1 (a j , b j ) .
Suppose further that for all z ∈ ∂R we have

F(z) • n(z) < 0 , (18)
where n(z) is the outer normal at z. Then the solution u(ξ, t) of ( 17) exists for all times t ≥ 0 and takes its values inside R.

Proof: This is Proposition 4.3 in Taylor [?], Chapter 15 (p.295).

Lemma 3 Let M t be a growing manifold. Suppose that the initial condition of the reactiondiffusion system (4) is in C M, R M and takes its values inside the rectangle

R = M j=1 (a j , b j ) .
Suppose further that for all (z, t) ∈ ∂R × [0, ∞) we have

F(z) • n(z) < c(t)n(z) • z ∀t ≥ 0, , (19)
where n(z) is the outer normal at z and c(t) is the growth rate. Then the solution u(t) of ( 4) is global and bounded, i.e. it exists for all times t ≥ 0 and takes its values inside R.

Proof: From Lemma 1 the reaction-diffusion system (4) can be transformed into

∂ t u = D∆ M u + G(t, u) ,
where G(t, u) := -nρ(t) ρ(t)u + ρ 2 (t)F(u). Taking a careful look at the proof of Lemma 2 in Taylor [?] we see that it also holds for nonlinearities that depend on time, provided

G(t, z) • n(z) < 0 for all (z, t) ∈ ∂R × [0, ∞), (20)
where n(z) is the outer normal at z. Therefore, since condition (20) is equivalent to hypothesis (19), we can apply Lemma 2 to obtain that the solution u(t) of ( 4) exists for all times t ≥ 0 and takes its values inside R.

Now we can conclude the proof of Theorem 4. If R = (-1, 1) M then n(z) • z = 1 for all z ∈ ∂R, which implies that (19) reduces to [START_REF] Gjorgjieva | Turing pattern dynamics for spatiotemporal models with growth and curvature[END_REF]. Therefore, using Lemma 3 we obtain the result.

Proof of Theorem 5

Let (M t ) 0≤t≤T be an isotropic growing manifold and define the fixed manifold S := M T . The linearisation of the operator on the fixed manifold S is

L S := D ρ 2 (T ) ∆ S + dF(0)
whilst the linearisation on the growing manifold (M t ) 0≤t≤T ) at time t = T is

L M T := D ρ 2 (T ) ∆ M T -c(T )I + dF(0) , c(T ) := n ρ(T ) ρ(T ) .
Since L S and L M T are second-order elliptic linear operators, their spectra consists on pure eigenvalues, i.e. there is no continuum spectrum for any of both. Let λ ∈ C be an eigenvalue on L S . Then there exists a non-trivial function φ : S → C solution of

L S φ = λφ.
The anti-blow-up effect of growth on pattern formation

Growth has an anti-blow-up effect because it enhances the possibility of global existence of solutions. Indeed, condition (5) in Theorem 4 is less restrictive than condition (18) in Lemma 2 because even if (18) does not hold (5) can be fulfilled. In that case, a solution of the system (17) on the fixed manifold M is perhaps only a time-local solution, but as a solution of the system (4) on the growing manifold (M t ) t≥0 it could be globally defined in time. In order to quantitatively assess this anti-blow-up effect we provided two examples on the scalar case, where there is blow-up for the corresponding equation. We found that on a growing manifold the blow-up occurs at a later time than on the fixed manifold, and if growth is fast enough then the blow-up does not occur at all.

The stabilising effect of growth on pattern formation

As it was shown in Theorem 5, growth shifts the spectrum towards the left in the complex plane by the explicit factor c(T ) > 0. This implies that the real parts of the eigenvalues are smaller on the growing manifold (M t ) 0≤t≤T at time t = T than on the corresponding fixed manifold S = M T , which is a gain of stability. It is worth to mention that our proof does not work for non-isotropic growing manifolds because we considered the growth rate to be independent of the space variable ξ. It would be interesting to see if the stabilising effect of growth holds for not only for non-isotropic growing manifolds property but also for more general growth regimes, e.g. then the growth factor is not exogenous but it is also one of the unknowns of the problem. From Theorem 5 we can also infer that growth is a regulatory mechanism for stability, in the sense that it enhances stability and selects the most stable patterns for expression. This fact is very important because it suggests that growth is an important factor in self-regulation features occurring in embryogenesis (see Meinhardt [START_REF] Meinhardt | Growth and patterning: dynamics of the stripe formation[END_REF]) and tumor growth (see Chaplain et al [START_REF] Chaplain | Spatio-temporal pattern formation on spherical surfaces: Numerical simulation and application to solid tumour growth[END_REF]). Whether these applications are possible is a crucial problem, which we would like to study in detail in the future.

Exponential growth factor

A very special type of growth factor is ρ(t) = ae rt because c(t) is constant if and only if the growth factor ρ(t) is of exponential type. This observation implies that the simplest case of growth to be added on a model is exponential, and therefore it is important to work on the exponential case before approaching a more general growth factor in order to gain some insight.

In that spirit we have shown that if r > 0 is big enough then the solutions of the system (4) are globally defined (i.e. there is no blow-up). But there are more features of the exponential growth. For example, Gjiorgjieva [START_REF] Gjorgjieva | Turing pattern dynamics for spatiotemporal models with growth and curvature[END_REF] showed that the system (4) on a 2-dimensional sphere with two morphogens and exponential isotropic growth has a constant equilibrium solution if and only if ρ(t) is exponential (see Lemma 5.1, p.50), and we can show that her result holds for any manifold and for any number of morphogens. Indeed, (4) has a constant equilibrium u 0 = (u 0 1 , . . . , u 0 M ) if and only if n ρ(t) ρ(t) u 0 = F(u 0 ) .

Therefore, if u 0 ≡ 0 then for any u 0 i = 0 we have

ρ(t) ρ(t) = F i (u 0 ) nu 0 i , (21)
which implies that ρ(t)/ρ(t) is constant, and in consequence ρ(t) is exponential. Therefore, if we have a constant equilibrium then the growth function is exponential and its growth rate is completely determined by the nonlinearity. In other words, whenever we find a constant equilibrium the growth factor is necessarily exponential, i.e. ρ(t) = ae rt , and we can calculate the growth exponent r > 0 using (21).

  , if F(u) and ρ(t) are C ∞ in their arguments then ∂ t u(ξ, t) is continuous in time, and in consequence u(ξ, t) is C 1 in time. Now, if we derivate (4) with respect to time we see that ∂ 2 t u(ξ, t) is continuous in time as well, and so u(t, x) is C 2 in time. Continuing this way it follows that u

Acknowledgements

I wish to thank Pablo Padilla at IIMAS -UNAM, Mexico, who encouraged me to look at the stabilizing effect of growth on pattern formation. I also wish to thank Miguel Angel Herrero at Universidad Complutense de Madrid, Spain, and Philip K. Maini at University of Oxford, UK, for the fruitful discussions we held during the preparation of this work.

Therefore, since L M T = L S -c(T )I it follows that φ : M T → C is a non-trivial solution of

which implies that λ -c(T ) is an eigenvalue of L M T .

Discussion

Reaction-diffusion systems on growing manifolds

We have shown here that the same results presented by Plaza et al [?] hold in the case of any n-dimensional manifold (in the sense of Definition 1). Moreover, the techniques we used to prove Theorem 1 are independent of the choice of an orthogonal parametrisation. This implies that one can choose the coordinate system that is better for explicit calculations, regardless if it is orthogonal or not.

Linear stability analysis

The following lemma summarizes the properties of the Laplace-Beltrami operator ∆ M .

Lemma 4 Let M be a manifold and consider the operator -∆ M . Then:

1. All eigenvalues of -∆ M are real and nonnegative.

2. Zero is an eigenvalue with multiplicity one.

All eigenspaces are finite dimensional.

4. There exists infinitely eigenvalues

and they accumulate only at infinity (i.e. theres is no finite accumulation point.

5. The eigenvectors of -∆ M constitute an orthonormal basis of L 2 (M).

All eigenvectors are smooth.

Proof: See Rosenberg [START_REF] Rosenberg | The Laplacian on a Riemannian manifold[END_REF], Theorems 1.29 (p.32) and 1.31 (p.35). Lemma 4 states that the Laplace-Beltrami operator ∆ M possesses the very same spectral properties than the Laplacian operator in euclidean, regular and bounded domains with Neumann boundary conditions. Therefore, all the linear stability analysis performed by Murray [START_REF] Murray | Mathematical Biology II: spatial models and biomedical applications[END_REF] can be applied to the case of a manifold, mutatis mutandis. Furthermore, the statement of Gjorgjieva and Jacobsen [START_REF] Gjiorgjieva | Turing patterns on growing spheres: the exponential case[END_REF] we quoted in the Introduction holds for any 2-dimensional manifold: growth increases the number of possible patterns but (generically) chooses lower eigenmodes. Indeed, let us assume the following conditions:

• The manifold M is 2-dimensional.

• We have only two morphogens, whose diffusion coefficients are different (say 1 and d > 1).

• The nonlinearity F(u) depends on a real parameter γ in the form

• The growth factor is exponential:

Under these conditions the system becomes

which is exactly the system ( 5)-( 6) in Gjorgjieva and Jacobsen [START_REF] Gjiorgjieva | Turing patterns on growing spheres: the exponential case[END_REF]. Moreover, if we substitute the spherical harmonics they use by the corresponding eigenvectors for ∆ M we can perform the same analysis they have already done, thus obtaining the same results for a general 2dimensional manifold. An open question we would like to address in the future is whether the linear stability analysis of Gjorgjieva and Jacobsen [START_REF] Gjiorgjieva | Turing patterns on growing spheres: the exponential case[END_REF] is also valid for n-dimensional manifolds with more general growth functions.

Qualitative properties of solutions

Whenever a pattern formation problem is addressed there are several "natural" questions related to the system (4). In this work we answered affirmatively the questions of existence, uniqueness and regularity, and we showed that growth shifts the eigenvalues of the system (4) towards the left in the complex plane (Theorem 5). We have also noticed that the linear stability results of Gjorgjieva and Jacobsen [START_REF] Gjiorgjieva | Turing patterns on growing spheres: the exponential case[END_REF] can be extrapolated to general surfaces. However, the bifurcation analysis is far from being complete, and symmetry breaking and asymptotic behavior for large times are open questions. We aim to study these properties in future works.


Stefan Buschhorn
email: stefan.buschhorn@ruhr-uni-bochum.de

Frank Brüssing

Radu Abrudan

Hartmut Zabel

Precessional damping of Fe magnetic moments in a FeNi film

We report on the element-resolved precessional dynamics of Fe magnetic moments in a homogeneous FeNi thin film. In our pump-probe experiment the magnetic system is excited by a magnetic field pulse from a stripline. The instantaneous response to the field pulse excitation is monitored as a function of time in a stroboscopic measurement using element selective X-ray Resonant Magnetic Scattering (XRMS). Our data show that Fe and Ni moments are aligned parallel to each other at all times, while they oscillate around the effective field direction given by the step field pulse and applied bias field. The field dependence of the precessional motion and damping of Fe magnetic moments is analyzed and compared to time-resolved Magneto-Optical Kerr Effect (tr-MOKE) measurement data from literature, showing good agreement. Additional studies proove the capability of our setup to conduct temperature dependent studies. In case of the presented FeNi system no changes in the frequency or damping behavior are observed within a temperature range of 150 -350 K.

Introduction

The studies of magnetization dynamics in ferromagnetic materials provide valuable information about fundamental material properties, which is complementary to or sometimes even more extensive than the information gained from magnetic hysteresis measurements. Whenever the mean magnetization experiences a torque induced by a non-collinear alignment between the effective field and the magnetization vector, a precessional motion around the effective field direction occurs and is accompanied by damping which forces the magnetization vector into the field direction [START_REF] Miltat | Magnets fast and small[END_REF][START_REF] Bauer | Suppression of magnetic-field pulseinduced magnetization precession by pulse tailoring[END_REF]. The damping of this precession is important for the speed of domain wall propagation and determines the timescale and stability of precessional switching at which future spintronic devices will be based on [START_REF] Th | Ultrafast precessional magnetization reversal by picosecond magnetic field pulse shaping[END_REF][START_REF] Tudosa | The ultimate speed of magnetic switching in granular recording media[END_REF]. In fact, both switching mechanisms, i.e., domain wall propagation and precessional switching, find their applications in different systems, like bit-patterned media [START_REF] Hellwig | Coercivity tuning in Co/Pd multilayer based bit patterned media[END_REF] or the race-track memory [START_REF] Parkin | Magnetic Domain-Wall Racetrack Memory[END_REF]. The eigen-frequency and the damping constant of a ferromagnetic layer can be determined from ferromagnetic resonance experiments (FMR) [START_REF] Farle | Ferromagnetic resonance of ultrathin metallic layers[END_REF]. However, in ferromagnetic alloys or multilayer systems conventional FMR does not allow to distinguish directly between the dynamics of the constituent elements. Therefore, FMR of magnetic heterostructures and multilayers may become difficult to analyze if several resonance lines are located very close. Often, the experimental results need to be supplemented by simulations where certain assumptions are made to model the underlying physical mechanisms and the resulting spectra are compared to the measured ones. This lack of element sensitivity has been overcome by performing time resolved Xray Magnetic Circular Dichrosim (XMCD) measurements while the sample is exposed to resonant microwave radiation, be it with a phase-locked [START_REF] Arena | Combined time-resolved xray magnetic circular dichroism and ferromagnetic resonance studies of magnetic alloys and multilayers (invited)[END_REF] or time-averaged [START_REF] Boero | Double-resonant x-ray and microwave absorption: Atomic spectroscopy of precessional orbital and spin dynamics[END_REF] technique. In addition to FMR-based techniques, pump-probe setups were realized in recent years. Time-resolved XMCD experiments in reflection or transmission geometry [START_REF] Bailey | Precessional dynamics of elemental moments in a ferromagnetic alloy[END_REF][START_REF] Martin | Layer resolved magnetization dynamics in interlayer exchange coupled Ni 81 Fe 19 /Ru/Co 90 Fe 10 by time resolved x-ray magnetic circular dichroism[END_REF] provide additional information about the constituting elements while using magnetic field pulse excitation. Each of the element-resolved experimental techniques has specific advantages and disadvantages, which have been described in some detail by Guan et al. [START_REF] Guan | Comparison of time-resolved x-ray magnetic circular dichroism measurements in reflection and transmission for layer-specific precessional dynamics measurements[END_REF]. We use time-resolved X-ray Resonant Magnetic Scattering (tr-XRMS) after pulse excitation [START_REF] Buschhorn | Time resolved x-ray resonant magnetic scattering in reflection geometry[END_REF]. This method allows to obtain structural information via resonant magnetic x-ray reflectivity measurements in combination with time-resolved dynamical experiments on one and the same sample under the very same sample conditions. Here we present results on a permalloy Fe 20 Ni 80 (Py) thin film. In recent years the precessional damping of Py has been investigated in much detail in pumpprobe experiments, including the thickness dependence [START_REF] Nibarger | Variation of magnetization and the Landé g factor with thickness in Ni-Fe films[END_REF], interlayer effects [START_REF] Th | Large-angle magnetization dynamics investigated by vector-resolved magnetization-induced optical second-harmonic generation[END_REF] and vector-resolved experiments [START_REF] Silva | Detection of coherent and incoherent spin dynamics during the magnetic switching process using vector-resolved nonlinear magneto-optics[END_REF]. Bailey et al. [START_REF] Bailey | Precessional dynamics of elemental moments in a ferromagnetic alloy[END_REF] have investigated the element-specific dynamics of Py, however the authors do not comment on the damping of the individual elements, which is of utmost importance for applications. We measured the element-resolved magnetization dynamics in a 25 nm Py film by a field-pump x-ray-probe experiment and studied the precessional damping of the Fe and Ni magnetic moments separately within the alloy as a function of applied field. Furthermore, we have investigated the temperature dependence of the damping constant.

Experimental procedures

For our pump-probe experiments we use a stripline geometry to generate a magnetic field pulse at the sample region. For reflection geometry experiments the stripline design is simpler than a coplanar waveguide structure used in FMR, and provides the possiblity to investigate the magnetization dynamics even in samples with very thick and complex layer structures or samples on a non-transparent substrate. A 7 × 7 mm 2 Si substrate supports a 350 µm wide stripline structure, consisting of a triple stack (Cu(75 nm)/Py(25 nm)/Cu(3 nm)) prepared by ion beam sputtering through a mask. The HV sputtering chamber has a base pressure of 2 × 10 -9 mbar and the Ar sputter gas has a partial pressure of 4 × 10 -4 mbar during deposition. The use of Si with a naturally formed oxide layer on its top prevents any substrate-induced in-plane anisotropy. The 75 nm thick Cu layer serves as main conducting layer, whereas the second Cu layer prevents the oxidation of the magnetic system. The stripline is connected to an electronic pulse generator providing voltage pulses of 10 V amplitude and 10 ns width to excite the magnetic system. The voltage pulse is directly converted into an Oersted field pulse H p that is perpendicular to the stripline. A bias field H b can be applied parallel to the stripline in order to align the initial magnetization prior to excitation.

For the experiments we used the ALICE chamber housing an x-ray diffractometer [START_REF] Grabis | Diffractometer for soft x-ray resonant magnetic scattering[END_REF] as the end station of the undulator beamline UE52-SGM at the Bessy synchrotron of the Helmholtz-Zentrum Berlin (HZB). The beamline provides photons with variable polarization and energy especially in the range covering the L 3,2 edges of the 3d transition metal elements. Therefore it is ideally suited for magnetic studies, be it absorption measurements, reflectivity studies [START_REF] Bergmann | X-ray resonant magnetic scattering study of [Co 2 MnGe/Au] n and [Co 2 MnGe/V] n multilayers[END_REF] or holography experiments [START_REF] Hellwig | Magnetic imaging with soft x-ray spectroholography[END_REF][START_REF] Günther | Microscopic reversal behavior of magnetically capped nanospheres[END_REF]. During single bunch operation mode the repetition frequency of the electron bunches is 1.25 MHz with a pulse length of 50 ps, enabling pump-probe type experiments for processes much faster than 800 ns, i.e. the time between two photon pulses. Temporal resolution is then achieved by electronically shifting the voltage pulse with respect to a trigger signal from the storage ring. With our system steps as small as 10 ps are possible which is well below the intrinsic probe pulse length. The reflected intensity is detected by a photodiode current and monitored as a function of delay time between pump and probe. In order to obtain reasonable statistics, we average over about 10 7 excitations, assuming that the dynamics of the system is repetitive and completely reversible.

In figure 1 a schematic of the experimental setup for time-resolved measurements is shown. The stripline is oriented perpendicular to the scattering plane along the ydirection, a pair of Helmholtz coils provides the bias field H b parallel to the stripline. The whole assembly together with hf wiring is vacuum compatible and mounted on the cold finger of a cryostat in the ALICE chamber. The bias field H b is used to saturate the sample and align the magnetization prior to excitation and thus defines the initial condition. An electromagnet used to apply a field parallel to the surface along the x-direction is part of the ALICE chamber and not shown explicitly in figure 1.

The diffractometer environment enables q-dependent studies, and any incident angle desired can be used. In general, the magnetic contribution to the scattered intensity is I m ∝ σ•M, where σ describes the photon helicity and M is the magnetization vector. In reflection geometry at small incident angles we therefore monitor the evolution of the in-plane magnetization component M x . The M z -or out-of-plane component is usually very small due to the strong in-plane anisotropy, and on top of that the sensitivity to this component is very weak . For these reasons also M z can be neglected, and the magnetic scattered intensity can be written as I m = I 0 m • sin Φ. Here I 0 m is the absolute magnetic signal at a given incident angle and photon energy, and Φ is the angle between the magnetization M and the y-axis.

One of the challenges for time-resolved experiments is to ensure the relative position between the sample and the beam, especially when cooling the sample. On the one hand, the feedback loop of the synchrotron might slightly change the beam with respect to the sample position. On the other hand, the sample sitting at the end of a long cold finger changes its position as a function of temperature. These experimental details are negligible for extended samples, but become important for narrow striplines and the more so the narrower the stripline is. For our case, with a 350 µm wide stripline we succeeded to observe a clear time resolved signal after a thorough temperature stabilization.

Figure 2.

Rocking scans with opposite field directions (I + and I -) to determine the asymmetry at each incident angle. The main graph shows the photocurrent as measured together with a horizontal line indicating the measuring range of the ammeter in order to obtain low-noise data in the incoming signal, the inset shows the normalized data to show that the difference between I + and I -is not an effect due to the ring decay.

Results and Discussion

Static experiments

For an optimized XRMS signal for both the Fe and Ni resonant edges, we first characterized the static magnetic properties of the Py film. Rocking scans where measured at various positions on the reflectivity curve to define a proper angle of incidence. Figure 2 shows these scans performed at various incident angles; a field of 275 Oe was applied along the x-axis to saturate the sample in the two opposite directions (I + ,I -), and the difference between the two signals is of magnetic origin. The main graph shows the photocurrent as measured, whereas in the inset these data are normalized to the ring current. A strong decay of the intensity is observed as the incident angle is increased, while the resulting magnetic asymmetry (I + -I -)/(I + + I -) is more or less the same with 27.8, 28.3 and 25.9 % at 5.0, 6.0 and 6.5 • , respectively. For angles larger than 6.5 • both the overall signal and the asymmetry are further reduced. For the ammeter used in this experiment, the horizontal line indicates the limit of the measurement range Figure 3. The upper panels show energy scans with circularly polarized photons at an incident angle of 6 • around the Fe and Ni L 3 edges, performed for two oppositely applied magnetic fields along the x-axis (see figure 1). The lower panels show the resulting magnetic asymmetry.

(2 × 10 -10 A). Below this level the background noise from the ammeter is almost two orders of magnitude smaller than for photocurrents exceeding this value. Therefore an angle of incidence of 6.0 • is the best compromise between a strong absolute and magnetic signal and a minimal detector noise level even for the highest photon flux immediately after an injection.

During alignment photon energy values close to the nominal absorption edges were used. To enhance the magnetic signal for the time-resolved experiments, the photon energy was then refined at the given incident angle. The stripline sample was saturated in-plane along the x-direction, and the photon energy was swept through the resonance while keeping the circular polarization constant. The result of these energy scans for positive and negative saturation (I + and I -, respectively) are shown for both elements in figure 3. The lower panels show the asymmetry which is the magnetic contrast at the respective energies. We concentrated on the L 3 edges, where the magnetic contrast is larger in comparison to the L 2 edges. In contrast to XMCD data, the energy scans carried out using XRMS contain both magnetic and charge contributions and therefore the shape strongly depends on the incident angle.

For a static characterization of the sample we measured hysteresis loops as shown in figure 4. As discussed above, the magnetic contribution solely depends on the angle Φ for a given incident angle and photon energy. For a hysteresis loop along the x-direction, positive and negative saturation are then equivalent to Φ = ±90 • , respectively. A net magnetization zero corresponds to an alignment along the y-axis, and is normalized to unity as this is the reference situation prior to any excitation. The detected intensity change during a time-resolved scan can thereby be converted into the opening angle Φ via the hysteresis loop data using I m = I 0 m • sin Φ. To the amplitude of the Oersted from the we xray recorded loops with and without pulsed excitation. Figure 4 shows data measured at the Fe edge: In both cases we have a coercivity of 4.5 Oe and only a weak anisotropy, but when the current pulse is introduced the whole hysteresis loop is shifted due to the additonal field H p along the x-axis. From this shift we can estimate the pulsed field to be in the order of +2.5 Oe, oriented parallel to the positive direction of the applied external field (i.e. in x-direction). Another method to quantify H p from the dynamic results is shown in the next section.

Time-resolved experiments

For the time-resolved experiments the initial magnetization is aligned with the bias field H b along the stripline (y-direction). There is no additional field applied in the x-direction. Upon the field pulse, we observe a time dependent change in intensity, implying that M x (x-component of the magnetization) changes in time. The intensity changes are then converted into opening angles Φ and plotted as a function of delay time to monitor the magnetization dynamics in the film.

To compare the magnetization dynamics of the Fe and Ni magnetic moments, we have plotted both signals in figure 5 for a set of bias fields. For each bias field, the photon energy was set to the Fe (full symbols) and Ni (open symbols) L 3 -edges, and we followed the intensity change as a function of time. All curves show an initial rise followed by a damped precessional oscillation, as expected for step-like field-pulse excitations. Two main features are clearly visible in this data set: with the increase of bias field, the opening angle decreases, and at the same time the frequency increases. The initial slope dφ dt is the same for all bias field values. This feature leads to a decrease in the response time, i.e. the time between the onset of the current pulse and the first transition through the new equilibrium direction. The data presented in figure 5 clearly show the same oscillation amplitude, frequency and phase for both Ni and Fe magnetic moments over the entire range of delay time. This is expected as the exchange coupling between the Fe and Ni moments is rather strong and confirms the observation by Bailey et al. as well [START_REF] Bailey | Precessional dynamics of elemental moments in a ferromagnetic alloy[END_REF]. The larger error bars for the Ni in comparison with the Fe data arise for two reasons: The asymmetry at the Ni edge is smaller by a factor of almost four, leading to an increase of the statistical errors in case of identical counting times. Second, for identical glancing angle the scattering vector for scanning the Ni edge is higher because of the shorter wavelength, implying that the reflectivity from the Py film is reduced.

In the following we concentrate on the data obtained at the Fe edge for further data analysis. We use a damped sinusoidal oscillation to describe the magnetic response following the field pulse:

Φ(t) = Φ 0 + Φ exp (-t/t 0 ) • sin(ωt + ϕ) ( 1 )
Here Φ 0 is the opening angle at equilibrium after the oscillations are damped out, and Φ is the oscillation amplitude. The precessional motion is described by the decay time t 0 and angular frequency ω = 2πf . Silva et al. have shown that this procedure gives results that are very similar to full LLG simulations [START_REF] Crawford | Subnanosecond magnetization dynamics measured by the second-harmonic magneto-optic kerr effect[END_REF]. In figure 6 the resulting fits are shown together with the data. The overall change in the magnetization direction assumes values of almost Φ = 30 • in the case of small bias fields. However, the precession itself is always restricted to ±2 • around the new equilibrium direction, thus the approximative description for small angle precession is appropriate. In FMR experiments, precessional cones exeeding few degrees are considered to be large and lead to the Suhl instability [START_REF] Anderson | Instability in the motion of ferromagnets at high microwave power levels[END_REF] where spin waves are supposed to be generated and reduce the absolute value of the magnetization. This is because in FMR experiments an rf-field is constantly applied to the sample leading to a forced precession of the sample magnetization. In a pumpprobe experiment, however, the excitation is given by a single field pulse which merely initiates a free precessional motion of magnetization. In similar MOKE experiments no generation of spin waves -manifesting themselves in a significant reduction of the absolute value of the magnetization -has been observed, despite excitation angles close to the switching regime [START_REF] Th | Large-angle magnetization dynamics investigated by vector-resolved magnetization-induced optical second-harmonic generation[END_REF]. Figure 5 shows that after about 4 ns the oscillations are damped out and the new equilibrium direction of the magnetization given by Φ 0 is reached. The latter one is given by minimizing the free energy of the system, and leads to the condition

H k • sin Φ 0 -H p + H b • tan Φ 0 = 0.
Even for the largest angle measured, we may assume sin Φ ≈ tan Φ within an acceptable error bar. In this way we can use the dependence of the opening angle on the bias field to again determine the pulsed field H p : figure 7a shows the change of the opening angle with varying bias field, where the solid line is a fit according to tan Φ = H p /(H b + H k ). H k accounts for the additional anisotropy along the stripline that is observed in hysteresis loops measured in both xand ydirections with MOKE. From the fit we obtain 2.4(±0.6) Oe for the magnitude of the pulsed field H p which is in very good agreement with what we deduced from our prior hysteresis shift test. The anisotropy field is 2.6(±0.2) Oe and presumably induced in the growth process. The anisotropy is in accordance with prior MOKE data.

In Py the saturation magnetization M s is much larger than H b + H k and therefore the frequency dependence can be well described by the Kittel formula

ω 2 = (γµ 0 ) 2 • [H eff (H eff + M s )] [23]
, where γ is the gyromagnetic ratio, µ 0 the vacuum permeability, and ω the angular frequency of the precession. In our case the effective field is a combination of the uniaxial anisotropy as calculated from the fit in figure 7a, and the applied and pulsed field at the sample. The calculations give H eff M s , and H p H b . Therefore the above equation can be simplified into

ω 2 = (γµ 0 ) 2 • [(H b + H k )M s ]
and can be used for a linear fit of the frequency dependence as shown in Figure 7b. Assuming a gyroscopic splitting factor g = 2 for Fe, we obtain M s = 619(±13) kA/m, which is in good agreement with prior SQUID measurements yielding M s = 570 kA/m. The anisotropy field deduced from the dynamic measurements is 4.5(±0.3) Oe, which is larger by a factor of 1.7 than the static result obtained from the opening angle. This difference can be attributed to an additional dynamic anisotropy contribution as described earlier [START_REF] Lopusnik | Different dynamic and static magnetic anisotropy in thin Permalloy films[END_REF]. In turn if we take the SQUID result as value for M s , the resulting splitting factor is g = 2.08(±0.04), which is in very good accordance with literature values [START_REF] Nibarger | Variation of magnetization and the Landé g factor with thickness in Ni-Fe films[END_REF]. From the fits in Figure 6, we also obtain a characteristic decay time t 0 for the damping of the precession. Figure 8 gives an overview on decay times determined from all pulsed measurements, with the blue horizontal line marking the averaged value. There is a tendency towards increased damping (i.e. smaller t 0 ) for small bias fields. This tendency has been pointed out by Nibarger et al. [START_REF] Nibarger | Damping as a function of pulsed field amplitude and bias field in thin film Permalloy[END_REF] who carried out tr-MOKE experiments with similar settings . The large error bars result from the small number of oscillations that can be recorded for our sample. Enhancing the number of oscillations should reduce the error bar, but is challenging to achieve with our setup: Increasing the bias field does not reduce the error bars, as the reduced signal amplitude does not qualitatively improve the data. On the hand, for the two smallest field the signal amplitudes are sizeable, but the decay times are short compared to the of one oscillation, which again does not improve the In general, we note the decay times from for elemental Fe moments in Py agree rather well with those obtained by measurements averaging over all constituting elements on this time scale.

Low temperature experiments

The ALICE chamber equipped a continuous flow cryostat which was operated with liquid Nitrogen (LN2) for our experiments. In Figure 9 we compare taken for two different bias fields at 150 K, 300 K, and 350 K. These data confirm that the necessary stability can be reached with our setup to conduct time-resolved scans at various temperatures. At the same time we do not observe significant changes in the angle, frequency and in the damping behavior within the temperature range 150-350 K. The increased error bars indicate the afore-mentioned challenges with the temperature The black line is the best fit to the 300 K data. For the results obtained at 150 K the onset of the pulse is hard to define and may lead to a small phase shift on the time-axis.

feedback loop. This result is in accordance with FMR results on Fe and Ni films in the high frequency regime obtained by Patton and Wilts [START_REF] Patton | Temperature dependence of the Ferromagnetic Resonance linewidth in thin NiFe films[END_REF], where the damping was found to be constant within the temperature range considered here, and with more recent FMR results on Py by Sierra et al. using a Vector Network Analyzer [START_REF] Sierra | Temperature dependent dynamic and static magnetic response in magnetic tunnel junctions with Permalloy layers[END_REF], which also match the low frequency range. In both references, an increase in damping is observed below about 120 K. Future experiments will focus on systems that significantly change their properties with temperature, e.g. a metal-insulator transition system adjacent to the Py layer. Then the additional effect of the metallic phase as a spin sink might be separated from a pure interface-related damping that is also present for the insulating phase.

Conclusion

We have observed element-resolved precessional dynamics in a 25 nm thick Fe 20 Ni 80 film shaped in form of a stripline with a width of 350 µm. A step-like field-pulse excitation in combination with pulsed circularly polarized x-rays is used for this pumpprobe experiment. Both Fe and Ni elemental moments precess at the same frequency and are aligned parallel. The damping determined from x-ray detected magnetization dynamics is in good accordance with MOKE results from literature averaging over both elements. The decay time is about 1.2 ns, with a tendency to decrease with decreasing precessional amplitude. The temperature dependence of the precessional dynamics was found to be negligible in the range 150 K to 350 K.

The presented data show the feasibility to conduct magnetization dynamics studies in thin films with an element-selective probe with a temporal resolution limited by the photon pulse length of 50 ps. In this way magnetic multilayer structures containing different magnetic elements can directly be accessed in the low-GHz regime. The shorter wavelength and larger penetration depth of soft x-rays in comparison to visible light used in MOKE experiments enable to also reach deeply buried layers, whereas the temporal resolution available with modern fs-Laser systems cannot be achieved with the synchrotron-based technique. The combination of the time-resolved setup with a diffractometer environment allows the observation at various scattering vectors, and therefore to separate interface and bulk magnetization properties. With the pumpprobe technique we access the regime of free precessional motion, which is not limited to small excitation angles and renders the investigation of large angle magnetization dynamics posssible, like it is the case in switching dynamics. Thus we circumvent the occurrence of the Suhl instability that limits typical resonant excitation experiments to a few degrees in opening angle [START_REF] Suhl | The theory of ferromagnetic resonance at high signal powers[END_REF]. Future experiments will focus on multilayer systems with emphasis on interlayer and interface effects.

Figure 1 .

 1 Figure1. Schematic of the setup and fields applied at the sample region. The 350 µm stripline is centered on a Si substrate and oriented perpendicular to the scattering plane. A pair of coils provides the bias field H b along the y-axis, the pulse field from the stripline is parallel to the x-axis. With our setup the change in the M x -component is measured while the magnetization precesses around the effective field direction H eff .

Figure 4 .

 4 Figure 4. Hysteresis loop measured at the Fe L 3 edge, with (open) and without (full symbols) applying a field pulse to the sample. A shift of 2.5 Oe is observed. The overall hard-axis behavior indicates a small anisotropy with its easy axis along the stripline direction.

Figure 5 .

 5 Figure 5. Comparison of the magnetization dynamics measured at the Fe (full) and Ni (open symbols) resonant edges for a set of different bias fields. The detected intensity is converted into opening angle Φ according to the hysteresis curves.

Figure 6 .

 6 Figure 6. Fit results for data obtained at the Fe edge using the damped sinusoidal function described in the text.

Figure 7 .

 7 Figure 7. In (a) the opening angle is shown as a function of bias field H b , the red line is a fit with the values H p and H k as free parameters, see text. In (b) the square of the frequency is plotted as a function of bias field H b , with a linear fit according to the Kittel formula.

H

Figure 8 .

 8 Figure 8. The full symbols show the damping constants from the fits to the data (figure 6) as a function of bias field. The open symbols show values obtained with a MOKE setup using a pulsed field of 3 Oe and are extracted from [21].

Figure 9 .

 9 Figure 9. Delay scans for two different bias fields and at three different temperatures.The black line is the best fit to the 300 K data. For the results obtained at 150 K the onset of the pulse is hard to define and may lead to a small phase shift on the time-axis.

  b (Oe) t 0 (ns) ∆t 0 (ns)

	2,1	1,07	0,13
	3,4	1,04	0,10
	6,9	1,20	0,15
	10,3	1,19	0,08
	13,7	1,34	0,13
	20,6	1,27	0,27
	27,4	1,39	0,20
	34,3	1,31	0,21

Acknowledgements

We would like to thank Denise Erb for her support during the measurements at Bessy II, Berlin, and Min-Sang Lee for valuable comments about the manuscript. This work was financially supported by the BMBF Verbundforschung (05KS7PC1), which is gratefully acknowledged. We are also thankful to the HZB Berlin for travel support under BMBF 05 ES3XBA/5.


Sarah Müller
email: sarah.mueller@ep5.rub.de

Dirk Luggenhölscher

Uwe Czarnetzki

Ignition of a nanosecond-pulsed near atmospheric pressure discharge in a narrow gap

The ignition phase and the transition to quasi DC glow operation of a narrow-gap near atmospheric pressure discharge in hydrogen are investigated experimentally. The discharge is ignited by a short 10 ns voltage pulse with a peak voltage of 1.3 kV followed by a 150 ns plateau of about 350 V. Pulsing is at 12 kHz which leaves a significant amount of residual charge between the individual pulses. Temporally resolved laser electric field measurement in the centre of the discharge employing a non-linear four-wave mixing scheme, ultra-high speed optical imaging by an ICCD camera at Balmer-alpha and Fulcher lines as well as the undispersed emission, and current and voltage measurements are performed. Special emphasis is put on a detailed analysis of the measured data by combining the results from the various diagnostics. This allows in addition to the directly measured quantities determination of the absolute evolution of the electron density, the development of space charge shielding, and the observation of the local electron dynamics. Pressure variations in a limited range indicate reasonable agreement with the Paschen law but raise also questions on the definition of the break down voltage under highly transient conditions.

Atmospheric pressure discharges under small spatial dimensions, so called microplasmas, have attracted large attention in recent years. On one hand large expectations on industrial applications exists while on the other hand the physics is often complex and difficult to investigate. In particular diagnostics is a challenge since most of the tools and techniques developed over decades for low-pressure, large scale discharges can not be applied here. Many types of microplasmas operate in pulsed mode and the transient phase of their ignition is even more difficult to access by adding to the already microscopic dimensions also fast temporal changes on ns or sub-ns timescales.

Here an experimental investigation of the rapid ignition phase and the subsequent transition to quasi-DC operation in a molecular gas (hydrogen) is reported. The geometry is one-dimensional to allow a simple interpretation. The main parameters determined directly are the discharge voltage, the current density, the electric field in the centre of the discharge, and the spatially and temporally resolved emission. First successful application of these diagnostics to a similar setup was reported in [1][2][3]. Here we go beyond by more detailed studies and in particular a careful analysis and interpretation of the data. From the combination of the directly measured quantities with simple and robust models further insight in the ignition dynamics is provided. The temporal evolution of absolute electron densities is obtained and spatial and temporal development of different groups of low and highly energetic electrons is observed.

Here a discharge ignited in a plane geometry is investigated (figure 2) that allows a simple one-dimensional interpretation of the obtained results. Two parallel plate electrodes of 3 cm diameter and 1.2 mm gap width made from stainless steel form the discharge volume. The power coupling is performed by repetitive nanosecond highvoltage pulses with a rise time of 10 ns, width of ∼ 10 ns and peak voltage of 1.3 kV. The high voltage pulse is followed by a quasi DC-phase of 350 V that lasts 150 ns. The repetition rate is 12 kHz. Short voltage pulses are generated from a DC power-supply by using a fast high-voltage switch (Behlke HTS-81). The discharge is operated between 200 mbar and 400 mbar of hydrogen. Simultaneous measurement of voltage and discharge current is performed by a high-voltage probe (Tektronix P6015A) and a current probe (Lecroy AP015) installed at the grounded electrode. The electric field in the centre of the discharge is measured by a laser spectroscopic technique similar to CARS [4][5][6]. It is based on recording the intensity of infrared (IR) radiation generated through anti-Stokes Raman scattering on the gas molecules in the investigated medium. The excitation scheme used for the field measurement is presented in figure 1. The process can be described as four-wave mixing based on the third order non-linear susceptibility that leads to generation of coherent radiation with field dependent intensity: The first two waves ω 1 and ω 2 are provided by two lasers (intensities I 1 and I 2 respectively) which are colinearly superimposed and focused. The third wave involved in this process (with ω 3 = 0) is effectively represented by the quasistatic electric field. The beam mixing leads to generation of a fourth wave (2.4 µm) whose intensity is proportional to the squared electric field strength. This scheme was originally invented by Ochkin et. al. [4][5][6]. Since the signal scales with the product of the green and red laser intensities the temporal resolution is slightly better than the individual laser pulse widths. We estimate a resolution (FWHM) of about 3 ns. In addition to the field dependent signal I IR , the anti-Stokes radiation (I CARS at 436 nm) is detected. Both signals depend on the laser intensities and the population density difference between the involved states in the rotational-vibrational energy system:

I IR = (N ground -N excited ) 2 I 1 I 2 |E| 2 (1)
I CARS = (N ground -N excited ) 2 I 2 1 I 2 (2)
By normalizing the static field dependent IR signal to the blue anti-Stokes signal, the dependence on the population distribution and the total gas density cancels out. Therefore, the electric field should linearly depend on the square root of the measured signal ratio:

|E| = C I 1 I IR I CARS (3)
The calibration constant C has to be determined experimentally. The laser intensity I 1 remains in the dependency and can be recorded in parallel. In the experiment, this beam is provided by a solid state laser which has a rather stable output (±3 %) so that I 1 is effectively a constant. In case of field inhomogenities along the propagation direction of the laser, the static field is the line average field. In any case, only the focal region contributes due to the non-linearity of the process. A schematic of the laser setup can be seen in figure 2. The laser system in this experiment consists of a frequency doubled injection seeded Nd:YAG laser at 532 nm (Continuum Powerlite Precision II) and a Nd:YAG-pumped dye laser at the Stokes wavelength 683 nm (Radiant Dyes Jaguar). Pulse energies used in the measurement are 30 mJ (Nd:YAG) and 5 mJ (Dye laser). Both laser beams are superposed linearly and focused by a plano-convex lens (f = 30 In order to obtain the calibration constant C, the intensities I 1 , I IR , and I CARS were detected while varying a static and homogeneous electric field without generation of a discharge. The calibration allowed also determination of the sensitivity. At 230 mbar, the minimum detectable field is 30 V/cm. With discharge operation the threshold is ten times higher due to electrical noise.

! ! ! "#$ %&' " $ " $ & ! ! ! ( )%&' * + ) # ,-. #/0 1. & 2 ! 3 !44 !-!
The spatial-temporal emission at Balmer-alpha (H α ), Fulcher lines, and the undispersed emission is detected by a fast ICCD camera (LaVision Picostar HR16). In this case quenching is beneficial since it reduces the lifetime of the excited states to values of the order of 10 ps [7]. The ICCD imaging is synchronized with current, voltage and electric field measurements. A series of images with a step size of 250 ps was taken during the discharge evolution. The gate time was set to 0.5 ns and the accumulation time was 2 s. Compared to the laser electric field measurement, the time resolution is much faster (0.5 ns vs. 3 ns). Due to the setup geometry the observation cone is shadowed by the electrodes (figure 3). In order to compensate for the shadowing effect a dimensionless factor describing the relative depletion can be calculated [START_REF] Preppernau | Glow discharge Spectroscopy[END_REF]:

B(y) = 1 - 1 π arccos y -y 1 -y 2 (4)
The variable y = x•L r•R is determined by the individual geometric dimensions. The factor B was taken into account in the image analysis.

The primarily measured data can be combined in a detailed analysis. In this analysis the mobility of electrons and ions is essential. Due to the high collisionality of the discharge, the ions are not able to move on the timescale of a few nanoseconds. The ion mobility µ Ion = 1.0 • 10 -3 m 2

Vs [START_REF] Macdaniel | The mobility and diffusion of ions in gases New York[END_REF] at p = 300 mbar taking into account the maximal electric field value of 800 V/mm, the distance the ions are able to move is below 8 µm. Therefore, the ions can be considered to be fixed at the location of their generation. However, the electrons have a much larger mobility (µ e = 0.13 m 2

Vs [START_REF] Raizer | Gas Discharge Physics[END_REF]). Under the same conditions as above, electrons therefore can transverse a distance of the order of the gap width. For the electron mobility the value reported by Raizer is used [START_REF] Raizer | Gas Discharge Physics[END_REF]. In the literature also other values are reported, e.g. [START_REF] Buckman | [END_REF], with differences in the absolute values of up to a factor of four. Further, the use of a constant value for the mobility is an approximation in itself and also field dependent mobilities are reported [12]. This uncertainty in the physical parameter sets limits to the absolute values derived in the subsequent data analysis. However, it should be noted that by the value used here at least consistency between different analysis approaches is achieved.

Figure 4 shows temporally resolved measurements of voltage, discharge current and electric field. The applied voltage rises within 15 ns to the peak value of 1.3 kV. At the same time, the electric field strength reaches its maximal value 800 V/mm. A current peak (I max = 11 A) is measured 4 ns later. The discharge cycle can be divided into two main phases: The ignition phase (10 < t < 35 ns) and a quasi-stationary phase (35 ns < t < 220 ns). When the voltage starts rising at 10 ns the electric field follows the applied voltage initially. At this early stage the space charge is still negligible but starts being developed. At 23 ns the gas breakdown occurs and the space charge starts shielding the applied electric field. During this time, the electric field is determined by the applied field reduced by the field produced by the total charge in the discharge volume. Assuming a homogeneous space charge, the reduced field can be expressed as

E sc (x, t) = V (t) d + Q(t)x 0 d • A (5)
where V (t) denotes the measured voltage, A the discharge area (here: 7.1 • 10 -4 m 2 ), and d the gap width (here: 1.2 mm). x = d 2 is the point of measurement. The charge Q is determined by the integrated current:

E sc (t) = V (t) d + 1 2 0 A t 0 I(t )dt (6)
This relation is represented by the early dotted line in figure 4. The simplified relation for the space charge reduced electric field is valid until about 30 ns. The curve fits the measured data already very well, although it was calculated based on a homogeneous density throughout the discharge gap. Since the electrons are drawn to the anode, the development of a density gradient pointing in this direction can be expected. This can be taken into account tentatively by introduction of a weighting parameter in the second term of equation 6. Assuming that 90% of the electrons are located between centre and anode, the calculated field decreases a few nanoseconds later than the dotted curve in figure 4 while the peak value deviates slightly more from the measured value. Overall, the difference is small. The breakdown voltage can in principle be determined from these measurement. However, the transient nature of the fast ignition makes a clear definition difficult. In figure 5 the full symbols represent the voltage where the measured field starts to deviate from the nominal field. The open symbols are the voltage reached at the maximum field strength. Further, one might also extrapolate the steeply rising current to zero and determine the voltage reached at this time of sudden current increase as the breakdown value. This gives even higher voltages than shown in figure 5. The results of figure 5 compare reasonably to the Paschen curve. As the secondary emission coefficient γ is not known exactly two curves representing the lower and upper limits of the probable parameter range are shown (γ = 0.1 and γ = 1).

Townsend's ionization coefficient α provides a convenient means to estimate the electron density increase out of the measured electric field strength. Neglecting spatial

⇒ n = n 0 exp    µ e t 0 α(E)Edt    . (8)
The first Townsend coefficient is given by the empirical formular α = Ap exp -Bp E [START_REF] Raizer | Gas Discharge Physics[END_REF]. The ionization coefficient together with the gas dependent constants A and B are taken from [START_REF] Raizer | Gas Discharge Physics[END_REF], Ap = 846.7 cm -1 and Bp = 22481.2 Vcm -1 at 230 mbar.

A density increase by factor of about 66 is determined (black diamonds in figure 6). The electron density rises rapidly in parallel with the measured current within an interval of only 8 ns. Subsequently, the density stays constant since the lower electric field strength is no longer sufficient for a significant ionization rate. The diffusion time in the afterglow is about 100 times longer than the time between successive pulses of τ = 8.3 • 10 -5 s. Since diffusion can therefore be neglected, the plasma is decaying by recombination. Due to the large cross section, initially produced H + 2 -ions are immediately converted to H + 3 [13][14][15]. The recombination rate of H + 3 with electrons is k = 2 • 10 -7 cm 3 /s [16]. The initial density is then independent of the final The absolute value of the plasma density can also be derived time resolved from the measured current and electric field strength. In this context it is important that ions do not move and charges are not lost by diffusion or recombination on the ns-timescale. The total current in the centre of the discharge is:

I = A • j = A(σE + 0 Ė) ( 9 )
Here the first term represents the conduction current with the conductivity σ and the second term the displacement current. Since the conductivity σ is density dependent, an expression providing the electron density time resolved during the whole discharge cycle can be found:

n(t) = 1 E(t)eµ e I(t) A -0 Ė(t) (10)
The initial density determined in this way is 4.5•10 16 m -3 (red squares in figure 6). This value compares remarkably well with the value estimated from the recombination. Also the rapid density development and transition to a quasi stationary value compares well to the analysis above based on the Townsend coefficient. The quasi stationary density is 2.5 • 10 18 m -3 , corresponding to an increase by a factor of 56 which is very close to the factor 66 derived above.

The constant density established in the quasi dc-phase is equivalent to a constant conductivity. In this phase, the electric field varies only slowly and the displacement current is negligible. Therefore, a direct proportionality between the current and the electric field exists. This proportionality is represented in figure 4 by the second dotted curve which connects the measured electric field data at later times. Excellent agreement is found throughout. At times earlier than 30 ns, the electric field varies rapidly, the displacement current becomes dominant, and the simple proportionality breaks down. Detailed insight into electron dynamics is provided by high-speed ICCDimaging. Spatially and temporally resolved measurement of visible radiation emitted by the discharge supports the time-resolved field measurements and their interpretation. In figure 7 the spatio-temporal evolution of the emission is shown. The imaging was performed unfiltered (figure 7a) and using interference filter to solely transmit H α -line (figure 7c) or the hydrogen Fulcher bands (figure 7b).

In agreement with current, voltage, and field measurements the ignition (23 ns -35 ns) and subsequent quasi DC-phase can be identified. In addition, an early (17 ns -23 ns), probably Townsend, regime becomes visible by very weak emission from the Fulcher lines (7b) and the undispersed measurement (7a) at the very early time of voltage rise. Here, an initial electron density remaining from the preceding discharge cycle is present and distributed probably homogeneously over the electrode gap since recombination leads to homogenization of the density distribution. At t = 10 ns, the voltage starts rising and the resulting electric field attracts the electrons to the anode. The pre-breakdown phase (17 ns -23 ns) is characterized by a relatively weak emission maximum stretching from 0.4 mm distance from the cathode to the anode (figure 7a). The electron density is still very low at this early stage. The dark region in front of the cathode can be interpreted as an ion space charge from where electrons have been removed. The electron drift distance can be calculated from the measured field by integrating the drift velocity u in time. Here it is important that space charge shielding is not yet effective and the field is homogeneous. With u = µ e • E it follows:

s = µ e 23 ns 10 ns Edt (11)
The result s = 0.4 mm matches the value found in the emission measurement very well. Since no atomic hydrogen is produced yet, emission in the H α -region is absent. With increasing field also ionization and electron density are rising. While the electron density increases by only a factor of about 60, the emission increases by more than three orders of magnitude in the ignition phase. Light emission starts to rise over the whole gap simultaneously. Since the voltage keeps rising, the breakdown condition is reached soon (t = 26 ns). At this time a very rapid ionization wave proceeds from anode to cathode [START_REF] Lagarkov | Ionization waves in electrical breakdown of gases[END_REF]. The tilting of the ionization front is hardly visible in the overview graph shown in figure 7. A zoom into the ignition interval is, therefore, provided in figure 8. Its propagation velocity can be roughly determined to 1•10 6 m/s. In parallel to the wave propagation a cathode sheath is developed. The wave slows down when approaching the edge of the sheath at a distance of 200 µm from the cathode. The H α emission intensity at this time peaks in the sheath with an amplitude of almost one order of magnitude higher than in the ionization wave. This is probably caused by massive secondary electron emission from the cathode. These secondary electrons will then gain much higher energies than the electrons in the bulk. Indeed, the emission at the Fulcher lines does not show the peak found at H α . This is consistent since the cross sections have a quite distinct energy dependence [START_REF] Janev | Post Elementary processes in hydrogen helium plasmas[END_REF][START_REF] Möhlmann | [END_REF][20]. While the atomic cross section has a late maximum at about 60 eV with a very slow decay at higher energies, the molecular cross section peaks already at 15 eV and decays quickly at higher energies. Consistently the Fulcher lines are more efficiently excited in the bulk when low electron energies are expected.

After the ionization wave has traveled to the cathode, the current keeps rising and produces a second emission maximum around 30 ns. Subsequently, together with the decreasing current, the emission intensity decreases to an absolute minimum (t = 40 ns). A sheath region in front of the cathode and a bulk region with less intense emission is generated. The emission in the quasi dc-phase is characterized by excitation via secondary electrons generated in the cathode sheath. Some weaker emission can also be identified close to the anode. The major part of the bulk is free of Balmer-alpha emission. Since it is unlikely that the atomic hydrogen generated in the ignition phase has vanished only nanoseconds later, this indicates the absence of energetic electrons. The Fulcher line sensitive to low energetic electrons still show some emission in this region with a peak also close to the anode. By considering the sheath edge to be located at the point of maximal emission, the sheath thickness can be determined to be 100 µm in the H α , and 125 µm in the Fulcher, and undispersed emission. The larger value is probably the more reliable one since in the Fulcher and the spectrally averaged measurement are more sensitive to lower energetic electrons. The density in the sheath can be estimated from the sheath width and the voltage balance across the gap U total = U s + U bulk . If the field in the bulk is assumed to be homogeneous, the measured electric field gives a bulk voltage drop of 163 V. With the total voltage 350 V, the sheath voltage can be determined to 187 V. Assuming a homogeneous density inside the sheath (homogeneous ionization and a negligible ion drift on this time scale) the density in the sheath can be calculated from Poisson's equation using the sheath width s = 125µm

n = 2U s 0 es 2 . (12)
A sheath density n = 1.3 • 10 18 m -3 results from this equation. This value is by a factor two smaller than the value determined in the bulk by evaluation of the current and field measurement, which seems to be reasonable. The discharge parameters in the quasi DC-phase can be compared to the tabulated literature values for normal DC discharges [START_REF] Raizer | Gas Discharge Physics[END_REF]. This gives a sheath voltage of 250 V (here 190 V), a current of 15 A (here 5 A) and sheath width of 50 µm (here 125 µm).

Although certain deviations are obvious, the order of magnitude values are very close.

In conclusion the measured data provide a detailed insight into the discharge and ignition dynamics. By applying some simple but robust analysis procedures further plasma parameters are inferred. In particular the temporal evolution of the absolute plasma density is revealed. These measured quantities and the parameters derived by the analysis procedure should provide a useful benchmark data set for simulation of discharge ignition at near atmospheric pressures.

Figure 1 .

 1 Figure 1. Excitation scheme relevant for CARS-based electric field measurement: (a) Generation of field-dependent IR-radiation. (b) Generation of field-independent anti-Stokes radiation.

Figure 2 .

 2 Figure 2. Experimental setup for the electric field measurements. Top right: optical setup for ICCD imaging.

Figure 3 .

 3 Figure 3. Shadowing of the observation angle occurs at the imaging of the discharge volume to the ICCD-chip. Here: L = 12 cm, r = 1.5 cm, R = 3 cm.

Figure 4 .

 4 Figure 4. Temporally resolved measurement of voltage, discharge current and electric field. The dotted curve shows the calculated electric field. The coloring indicates the beginning of the quasi-stationary phase of the discharge evolution at about 35 ns, while the ignition phase is uncoloured.

Figure 5 .

 5 Figure 5. Breakdown voltage resulting from the field measurements. Full symbols denoted the reading when the electric fields starts to deviate from the applied voltage, the open symbols result from the measured peak values.

Figure 6 .

 6 Figure 6. Black diamonds and right axis: density increase during the ignition process calculated from the field dependent Townsend ionization rate (equation 8). Red squares and left axis: Time evolution of the electron density derived from the current and electric field measurement (equation 10). Scaling of the right y-axis was selected to match the absolute values on the left axis.

Figure 7 .

 7 Figure 7. Spatially and temporally resolved and radially integrated measurement of visible radiation emitted by a nanosecond pulsed discharge and simultaneous measurement of electric field [kV/mm], applied voltage [kV], and plasma current [10 A]. (a) Unfiltered emission. (b) With interference filter transmitting between 600 nm and 625 nm (Fulcher-series). (c) H α -line recorded with interference filter.

Figure 8 .

 8 Figure 8. Zoom into the ignition phase. Example here: Spectrally averaged measurement. The arrow indicates the ionization wave traveling from the anode to the cathode with a velocity of about 10 6 m/s.

Acknowledgments

This work was supported by the 'German Research Foundation' (DFG) in the frame of 'Research Unit 1123 -Physics of Microplasmas', the Research Department 'Plasmas with Complex Interactions' of the Ruhr-University Bochum, and the Research School of the Ruhr-University Bochum. The authors gratefully thank Igor Adamovich, Zoltan Donko, Julian Schulze, and Satoshi Hamaguchi for fruitful discussions.