Arturo Kohatsu-Higa

Antoine Lejay

Kazuhiro Yasuda

On Weak Approximation of Stochastic Differential Equations with Discontinuous Drift Coefficient 1

In this paper, weak approximations of multi-dimensional stochastic differential equations with discontinuous drift coefficients are considered. Here as the approximated process, the Euler-Maruyama approximation of SDEs with approximated drift coefficients is used, and we provide a rate of weak convergence of them. Finally we present a rate of weak convergence of the Euler-Maruyama approximation of the original SDEs with constant diffusion coefficients.

Introduction

In mathematical finance, one describes asset price processes as the solution to the following stochastic differential equations (SDEs):

dX t = b(t, X t ) dt + σ(t, X t ) dW t . (1.1)
where b and σ are certain functions and W t is a Brownian motion. Then we consider a function f , which represents a payoff function in financial derivatives, and one write its associated option price as the expectation E[ f (X T )], where T is a maturity of the option and X T is the asset price at T . Note that we are using the interpretation of the expectation using a financial situation, but, of course, it is also important in many other fields and applications.

It is rare the occasion when one is able to calculate the previous expectation analytically. Therefore in order to obtain its value, one resorts to computer simulations and tries to obtain an approximated value. In practice, two kinds of approximations are needed to simulate this expectation. One is an approximation of the SDEs (1.1) and the other is an approximation of the expectation. For the latter, one can typically use the Monte-Carlo method, which is based on law of large numbers in probability theory. On the other hand, for the former, the Euler-Maruyama approximation is often used. The Euler-Maruyama approximation can be described as follows: For simplicity, we split the interval [0, T ] equally in n subintervals and let the length of each time subinterval ∆t be equal to T n , X0 = x, Xi+1 = Xi + b(i∆t, Xi )∆t + σ(i∆t, Xi )

√ ∆tξ i ,
where the random variables ξ i , i = 0, 1, • • • , n -1, are independent of each other and are distributed according to a N(0, I d ) law, where 0 is the d-dimensional zero vector and I d is d × d-unit matrix. When we approximate stochastic processes, one needs a criteria in order to determine the quality of the approximation. One mainly uses the following two criteria (strong error and weak error): the definition of an approximation with strong error of order γ > 0 is that there exists a positive constant C, which does not depend on ∆t, such that

E X T -Xn ≤ C∆t γ .
Under enough regularity for coefficients b and σ, the strong error has the order 1/2 for the above Euler-Maruyama approximation. For more details, readers can refer Exercise 9.6.3 in Kloeden and Platen [START_REF] Kloeden | Numerical Solution of Stochastic Differential Equations[END_REF]. The definition of weak error with order γ > 0 is that for all functions f in a certain class, there exists a positive constant C, which does not depend on ∆t, such that

E f (X T ) -E f Xn ≤ C∆t γ .
Here under enough regularity on the coefficients σ and b and on f , we have the weak error with order 1 for the Euler-Maruyama approximation. The purpose of this paper is to treat an SDE with discontinuous drift coefficients and obtain an order of weak error for its approximation. Precisely speaking, we consider an SDE with an approximated drift coefficient b ǫ , which is approximated using the Euler-Maruyama approximation. Then, one uses the approximated process as the approximation of the original SDEs. Then we estimate an order of the weak error between the original SDEs and the approximated process. In the latter part of this article, we deal with an SDE with constant diffusion coefficients and obtain an order of the weak error between the SDEs and their approximated process to which the Euler-Maruyama approximation is directly applied.

SDEs with discontinuous drift coefficients are of course used in various fields. For instant, in mathematical finance, if one wants to model a stock price process whose trend dramatically changes when a factor goes down a threshold value. In this case, the drift can be modeled as taking two values specified by some indicator function. This kind of SDE also appears in some control problems.

Weak error of SDEs with discontinuous coefficients (not only drift coefficients, but also diffusion coefficients) have been studied in Chan and Stramer [START_REF] Chan | Weak Consistency of the Euler Method for Numerically Solving Stochastic Differential Equations with Discontinuous Coefficients[END_REF] and Yan [START_REF] Yan | The Euler Scheme with Irregular Coefficients[END_REF]. However in their papers, they only proved weak convergence of the Euler-Maruyama approximation, not mentioned an order of the weak convergence. And also strong error and the rate are studied in Przybylowicz [START_REF] Przybylowicz | The Optimality of Euler-type Algorithms for Approximation of Stochastic Differential Equations with Discontinuous Coefficients[END_REF] for SDEs with some type of discontinuous coefficients. Note that in this paper, the diffusion coefficients of our SDEs have enough regularity.

This paper is organized as follows: Some notations and assumptions are given in Section 2. We provide our main result on a rate of weak errors under SDEs with discontinuous drift and nonlinear diffusion coefficient in Section 3, and also give results under constant diffusion coefficients in Section 4. Finally we give some numerical results in Section 5. Proofs of theorems and so on below can be found in Kohatsu-Higa, Lejay and Yasuda [START_REF] Kohatsu-Higa | Approximation methods for stochastic differential equations with non-regular drift[END_REF].

Notations and Hypotheses

Let d ∈ N. The space of continuous functions that are slowly increasing is denoted by

C S l (R d ). A function f in C S l (R d ) is such that for every k > 0, lim |x|→∞ | f (x)|e -k|x| 2 = 0. Fix T > 0. Let H be the set [0, T ) × R d and H = [0, T ] × R d .
Let σ be a measurable function on [0, T ] × R d with values in the space of symmetric d × d-matrices. We set a = σσ * and assume that there exist some positive constants Λ and λ (Λ ≥ λ > 0) such that λ|ξ| 2 ≤ ξ * a(t, x)ξ ≤ Λ|ξ| 2 , for all (t, x) ∈ H, and all ξ ∈ R d , (H1)

σ is uniformly continuous on H. ( H2
)
Remark 2.1 Note that (H1) gives a lower and upper bound on the eigenvalues of a, which are from the very construction equal to the eigenvalues of σ (we have chosen σ to be symmetric) for which (H1) holds with λ and Λ replaced by √ λ and √ Λ.

Let us also consider a measurable function

b from [0, T ] × R d to R d such that |b(t, x)| ≤ Λ for all (t, x) ∈ H. ( H3
)
From now on, we always assume (H1), (H2) and (H3) for b and σ. Now, we give some notations. Fix α > 0. Let H α (R d ) be the space of continuous, bounded functions with continuous, bounded derivatives up to order ⌊α⌋ and such that ∂ ⌊α⌋ x f is (α-⌊α⌋)-Hölder continuous. Let H α/2,α (H) be the set of continuous functions with continuous derivatives ∂ r t ∂ s x u for all 2r + s < α and such that

u H α/2,α = 2r+s≤⌊α⌋ sup (t,x)∈H |∂ r t ∂ s x u(t, x)| + 2r+s=⌊α⌋ sup (t,x),(t,y)∈H |∂ r t ∂ s x u(t, x) -∂ r t ∂ s x u(t, y)| |x -y| α-⌊α⌋ + 0<α-2r-s<2 sup (t,x),(v,x)∈H |∂ r t ∂ s x u(t, x) -∂ r t ∂ s x u(v, x)| |t -v| (α-2r-s)/2
is finite.

Main Theorems

Let σ and b satisfy (H1)-(H3). These conditions are sufficient to ensure the existence of a unique weak solution (X, (F t ) t≥0 , P x ) to

X t = x + t 0 σ(s, X s ) dB s + t 0 b(s, X s ) ds (3.1)
for a Brownian motion B.

Remark 3.1 If X t = x + t 0 σ(s, X s
) dB s has a strong solution, then (3.1) also admits a strong solution (See Veretennikov [START_REF] Veretennikov | On strong solutions and explicit formulas for solutions of stochastic integral equations[END_REF]).

Let b ǫ be a family of measurable coefficients on H with |b ǫ (t, x)| ≤ Λ for (t, x) ∈ H. Let us consider the unique weak solution (X ǫ , (F t ) t≥0 , P x ) to

X ǫ t = x + t 0 σ s, X ǫ s dB s + t 0 b ǫ s, X ǫ s ds. (3.2)
Since b ǫ and b are bounded, the distribution of X ǫ may be deduced from the distribution of X through a Girsanov transform.

For T > 0, let X ǫ be the continuous solution of the Euler-Maruyama scheme of step size

T/n. If φ(s) = sup{t ≤ s | t = k/n for k ∈ N}, then X ǫ t = x + t 0 σ φ(s), X ǫ φ(s) dB s + t 0 b ǫ φ(s), X ǫ φ(s) ds. (3.3)
When σ and b ǫ belong to an appropiate class of functions M (for example M = H α/2,α (H) for some α > 0 or M = C 1,3 b (H)), and when f belongs to a proper class of functions F (for example,

F = H 2+α (R d ) or F = C 3 (R d ) ∩ C S l (R d ))
, a rate of weak convergence of the Euler-Maruyama scheme X ǫ is known. This means that there exists some constant C ǫ such that

E f X ǫ T -E f X ǫ T ≤ C ǫ n δ . Assume that C ǫ = O(ǫ -β
). This is in general the case when one chooses to use a regularization b ǫ of b by using mollifiers.

On the other hand, as we will show below in Proposition 3.2 and Remarks 3.3 and 3.5, one has

E f (X T ) -E f X ǫ T ≤ C ′ E        T 0 |b(s, Y s ) -b ǫ (s, Y s )| p ds q/p        1/q (3.4)
for some appropriate values of p and q and positive constant C ′ . Assume that the quantity in the right-hand side of (3.4) decreases to 0 as O(ǫ γ ). Optimizing over the choice of ǫ leads to the following theorem.

Assume that f belongs to some appropriate class of functions F, and an approximation b ǫ of the drift b belongs to some class of functions M in a way such that

E f (X T ) -E f X ǫ T = O(ǫ γ ) (3.5)
and

E f X ǫ T -E f X ǫ T = O 1 ǫ β n δ .
(3.6)

Then for ǫ = O(n -δ/(γ+β) ), E f (X T ) -E f X ǫ T ≤ O(n -κ )where κ = δγ γ + β .
Under the assumptions (3.5) and (3.6), we have the order κ of the weak error among the SDEs (3.1) and the approximated process (3.3). Therefore, from now on, our interest is to find some conditions that the assumptions (3.5) and (3.6) hold.

A Perturbation Formula

Through Theorem 3.2 and the remarks below, we can find some situations where Assumption (3.5) holds.

Let X be the solution to (3.1) and X ǫ be the solution to (3.2).

Theorem 3.2 For α > 2 and p > 2 such that 1/α + 1/p < 1/2 and f ∈ C S l (R d ),

E f (X T ) -E f X ǫ T ≤ C 2 (α, p, T )A T (ǫ) Var P ( f (X T )) with C 2 (α, p, T ) = T 1/2-1/p exp          T Λ 2 λ -1          α - 1 2 + 1 - 2 α α 1 2 + 1 p -1 α 1 2 -1 p -1                   , A T (ǫ) = E 0 T 0 |b ǫ (s, Y s ) -b(s, Y s )| p ds 1/
for all (t, x, y) ∈ R + × R d × R d . Then for any 1 < r, q ≤ +∞ satisfying d/2r + 1/q < 1, it follows that

A T (ǫ) ≤ C 3       T 0 R d |b(s, y) -b ǫ (s, y)| pq dy r/q ds       1/rp ≤ C 3 b -b ǫ L rp,qp (H) ,
where for r < +∞ set

f L r,q (H) =        T 0 T 0 | f (s, x)| q dx r/q ds        1/r
, and also set f

L ∞,q (H) = sup t∈[0,T ] f (t, •) L q .
Remark 3.4 Such estimate (3.7) holds for example if the diffusion coefficient a belongs to H α/2,α (H) for some α > 0 (See for example Ladyženskaja [7, § IV.13, p. 377]).

Remark 3.5 Even in absence of a Gaussian upper bounds, the Krylov estimate (Krylov [START_REF] Krylov | An inequality in the theory of stochastic integrals[END_REF] or Bass [1, Theorem 7.6.2, p. 114]) could also be used with Hypothesis (H1) in order to get an estimate on A T (ǫ). In this case of a homogeneous coefficient b, from the Krylov estimate, we have

|A T (ǫ)| ≤ C(λ, Λ)e T b -b ǫ L d p .
In case of a time-inhomogeneous coefficient, a similar estimate could be obtained but on the bounded domain case and one should then estimate the exit time from such domains.

Rates of Convergence of the Euler-Maruyama Approximation with Regular Enough Coefficients

We now exhibit some situations where Assumption (3.6) holds, under the weakest possible assumptions on the regularity of the coefficients. Note that other results may hold (See Theorem 4.3 below).

Case of Hölder continuous coefficients

The weak rate of convergence of the Euler scheme when the coefficients of the PDE are Hölder continuous has been studied by R. Mikulevicius and E. Platen [START_REF] Mikulevičius | Rate of convergence of the Euler approximation for diffusion processes[END_REF].

Theorem 3.6 (R. Mikulevicius and E. Platen [START_REF] Mikulevičius | Rate of convergence of the Euler approximation for diffusion processes[END_REF]) If for α ∈ (0, 1) ∪ (1, 2) ∪ (2, 3), b and a belongs to H α/2,α (H) and f ∈ H 2+α (R d ), then there exists a constant K such that

E f (X T ) -E f X T ≤ K n E(α) with E(α) =              α/2 if α ∈ (0, 1), 1/(3 -α) if α ∈ (1, 2), 1 if α ∈ (2, 3).
Besides, the constant K is linear in b H α/2,α and a H α/2,α .

Case of smooth coefficients

Theorem 3.6 requires the coefficients to be Hölder continuous. Of course, the convergence rate is better for smooth coefficients. But in order to achieve a rate equal to 1, it requires a to be in H α/2,α (H) with α > 2 and a terminal condition in H 2+α (R d ) and then with a better regularity than C 4 p . With a bit more regularity on a and b (if we use molifier for the approximation, b ε has enough regularity), we see that we achieve a convergence rate equal to 1 provided that f in only in C 3 (R d ) ∩ C S l (R d ) by using Malliavin calculus.

Theorem 3.7 Assume that f in C 3 (R d ) ∩ C S l (R d ), b ǫ ∈ C 1,3 b (H) and σ ∈ C 1,3 b (H). Then for a uniform step size T/n, E f (X ǫ T ) -E f X ǫ T ≤ C n b ǫ 3,∞ ,
where C is some positive constant and b ǫ 3,∞ is defined as follows;

b ǫ 3,∞ = 3 j=0 ∂ j b ǫ ∂x j ∞ .

Example

b ǫ (x) =                        0, (-∞, ζ 1 -2ǫ) ∪ (ζ 2 + 2ǫ, ∞), 1 2ǫ x - ζ 1 -2ǫ 2ǫ , [ζ 1 -2ǫ, ζ 1 ), - 1 2ǫ x + ζ 2 + 2ǫ 2ǫ , (ζ 2 , ζ 2 + 2ǫ], 1, [ζ 1 , ζ 2 ].
Then we have the following orders:

for p > 2, ∞ -∞ |b ǫ (x) -b(x)| p dx 1 p = 4ǫ p + 1 1 p = O ǫ 1 p . (3.8)
And the rate of the divergence of b ǫ H α is ǫ -1 .

If we use a mollifier with the Gaussian kernel as b ǫ :

b ǫ (x) = ∞ -∞ b x -u ǫ 1 √ 2πǫ exp - u 2 2ǫ 2 du,
then we have the same order of the convergence as the above (3.8) and this b ǫ has enough regularity. And also the rate of the divergence of b ǫ 3,∞ is ǫ -3 .

Constant Diffusion Case

We now consider a simple case of a time-homogeneous coefficient and a constant diffusion coefficient. To keep it simple, we assume that σ is the identity matrix and then that X is solution to

X t = x + B t + t 0 b(X s ) ds (4.1)
for a Brownian motion B with distribution P. Let b ǫ be a family of approximations of b satisfying (H3). Let X and X ǫ be the continuous Euler-Maruyama schemes

X t = x + B t + t 0 b X φ(s) ds and X ǫ t = x + B t + t 0 b ǫ X ǫ φ(s) ds. Lemma 4.1 For p > 2, there exists a constant C 3 (p, Λ, T ) such that E f X T -E f X ǫ T ≤ C 3 (p, Λ, T ) Var( f (x + B T )) b -b ǫ L p .
The next lemma is a direct consequence of Theorem 3.2 and the Hölder inequality of the Gaussian density.

Lemma 4.2 For p > d ∨ 2, there exists a constant C 4 (p, Λ, T ) such that E f (X T ) -E f X ǫ T ≤ C 4 (p, Λ, T ) Var( f (x + B T )) b -b ǫ L p .
The rate of weak convergence of the Euler-Maruyama scheme to the solution to (4.1) has been studied by V. Mackevičius in [START_REF] Mackevičius | On the convergence rate of Euler scheme for SDE with Lipschitz drift and constant diffusion[END_REF] for a drift coefficient which is Lipschitz continuous. The proof is given for the dimension d = 1, but it is remarked in the article that it is suitable whatever the dimension (See Remark below Theorem 1 in [START_REF] Mackevičius | On the convergence rate of Euler scheme for SDE with Lipschitz drift and constant diffusion[END_REF]).

Let us denote by C 3 p (R d ) the space of functions on R d that are three times continuously differentiable with all the derivatives up to order 3 of polynomial growth. Of course,

C 3 p (R d ) ⊂ C S l (R d ).
E f X ǫ T -E f X ǫ T ≤ C 5 (T, Λ, f ) n Lip(b ǫ ).
E f (X T ) -E f X T = O n -d p+d .
Remark 4. [START_REF] Krylov | An inequality in the theory of stochastic integrals[END_REF] We see that the rate of weak error converges to 1/2 (resp. 1/3) when d > 2 (resp. d = 1) when p → d (resp. p → 2). However, the constants hidden in the O(n -d/(p+d) ) explode to infinity as p → d ∨ 2. This means that with our estimates, a better rate of convergence is obtained at the cost of a bigger constant in front of the rate.

Remark 4.7 In the proof of Theorem 4.5, we choose an optimal size of ǫ as O(n -p p+d ). Then we obtain the above rate of the weak error.

Numerical Results

In this section, we give some preliminary numerical experiments in order to determine if the rates of weak convergence are optimal and to which extent the slower rate of convergence can be observed. Here we consider the following SDE:

X t = x + t 0 b(X s ) ds + W t , (5.1)
where

b(x) = θ 1 , x ≤ 0, θ 0 , x > 0.
This process is called a Brownian motion with two-valued, state-dependent drift, which is related to a stochastic control problem. Then from Karatzas and Shreve [3, Section 6.5], the transition density function is given as follows:

p t (x, z) =                          2 ∞ 0 t 0 e 2bθ 1 h(t -s; y -z, -θ 1 )h(s; x + y, -θ 0 ) ds dy, x ≥ 0, z ≤ 0, 2 ∞ 0 t 0 e 2(bθ 1 +zθ 0 ) h(t -s; y, -θ 1 )h(s; x + y + z, -θ 0 ) ds dy + 1 √ 2πt exp - (x -z + θ 0 t) 2 2 -exp - (x + z -θ 0 t) 2 2 -2θ 0 x , x ≥ 0, z > 0,
where set

h(t; x, µ) = |x| √ 2πt 3 exp - (x -µt) 2 2t , t > 0, x 0, µ ∈ R.
Note that if θ 1 = -θ 0 = θ > 0 and x = 0, the distribution of X t is symmetric with respect to y-axis. So that when f is an odd function, we have E[ f (X t )] = 0.

Two approximated processes are attempted: one is the Euler-Maruyama approximation of the original SDE (5.1), and the other is the Euler-Maruyama approximaton of SDE with the approximated drift coefficient

b ǫ (x) =              θ 1 , x ≤ -ǫ, θ 0 -θ 1 2ǫ x + θ 0 + θ 1 2 , -ǫ < x ≤ ǫ, θ 0 , x > ǫ, for ǫ > 0. From Remark 4.7, set ǫ = n -2 3
, where n is a number of time steps of the Euler-Maruyama approximation.

Case

: θ 1 = -θ 0 = 1 and f (x) = x
In this section, we show a numerical result in the case of θ 1 = -θ 0 = 1, f (x) = x and the initial value X 0 = 0. Then the true value of E[ f (X 1 )] = 0 since f (x) = x is an odd function.

Through Figure 1 to Figure 3, x-axis denotes the number of time steps n until time 1 from 10 to 10 3 with logarithmic scale. Weak errors of simulation results are reported at a logarithmic scale on the y-axis, that is |E[ f (X 1 )] -E[ f (X 1 )]| (thin line) and |E[ f (X 1 )] -E[ f (X ǫ 1 )]| (dotted line), where to obtain their expectation values, we use the Monte-Carlo method with 10 7 simulations for each n. If they are parallel to the thick straight line, the convergence rate has the order 1.

The numerical result in the case of f (x) = x is the following: From Figure 1, it is easy to find that the convergence rate of the Euler-Maruyama method has order 1, but for the Euler-Maruyama method with the approximated drift, the approximation converges much faster than the uncorrected one. We have tested three cases above, the weak convergence rate of the Euler-Maruyama approximation in all of them is 1. And in the case of the Euler-Maruyama approximation with the approximated drift, we could not obtain the rate of convergence because the approximation converges too fast for f (x) = x and 1(x > 0) -1(x ≤ 0), but for f (x) = x 2 , we find that the convergence rate is 1. This is probably due to how ǫ is chosen. In this case, we have chosen this example because we can obtain the weak limit in closed form. In order to have slower orders, we need to consider more complicated situations.

  Here we provide an example of order of ǫ in the case of the indicator function b(t, x) = 1 [ζ 1 ,ζ 2 ] (x) for x ∈ R and ζ 1 < ζ 2 . If we use the following b ǫ for an approximation of b, b ǫ has the Lipschitz continuity: for ǫ > 0,

Theorem 4 . 3 (

 43 R. Mackevičius, [8, Theorem 1]) If b ǫ is bounded Lipschitz continuous with constant Lip(b ǫ ) and f ∈ C 3 p (R d ), then there exists a constant C 5 (T, Λ, f ) such that

Remark 4 . 4

 44 The statement of Theorem 1 in Mackevičius[START_REF] Mackevičius | On the convergence rate of Euler scheme for SDE with Lipschitz drift and constant diffusion[END_REF] is slightly different since b is not assumed to be bounded. Yet it is clear from the proof that the constant is linear in Lip(b ǫ ) if b is also bounded. For a set G in R d , we define G(ǫ) = {x ∈ R d |d(x, G) ≤ ǫ}, where d(x, G) = inf y∈G |x -y| is the distance between x and G.

Theorem 4 . 5

 45 Let b be a bounded function on R d which is Lipschitz except on a set G such that the Lebesgue meas(G(ǫ)) = O(ǫ d ). Then for any f ∈ C 3 p (R) and p > d ∨ 2,

Figure 1 :

 1 Figure 1: No. of time steps -weak error ( f (x) = x).

Figure 3 :

 3 Figure 3: No. of time steps -weak error ( f(x) = 1(x > 0) -1(x ≤ 0)).

This paper is an abbreviated and preliminary version of A. Kohatsu-Higa, A. Lejay and K. Yasuda[START_REF] Kohatsu-Higa | Approximation methods for stochastic differential equations with non-regular drift[END_REF].

Case: θ

Here we use the same values of parameters in the previous section and let f (x) = x 2 . From Karatzas and Shreve [3, Exercise 6.5.3, pp.441], we have

where set

And in the case of x = 0 and t = 1, we obtain E[ f (X 1 )] = 0.333369. The numerical result in the case of f (x) = x 2 is the following: From Figure 2, we easily find that the rate of convergence in the both methods is 1.

Case:

In this section, we use f (x) = 1(x > 0) -1(x ≤ 0) which does not have regurality and does not belong to our theorem. Note that the funciton f is symmetric with respect to the origin a.e. and X t has the continuous and symmetric density function so that we have E[ f (X 1 )] = 0. The numerical result in the case of f (x) = 1(x > 0) -1(x ≤ 0) is the following: From Figure 3, it is easy to find that the convergence rate of the Euler-Maruyama method has order 1, but as before the Euler-Maruyama method with the approximated drift, converges faster.


Marco ; Zanatta
email: zanatta@science.unitn.it

C Armellini

Aldo Fontana

F Rossi

Temperature evolution of the boson peak and Debye scaling in vitreous B 2 O 3

Keywords: glass, boson peak, Raman spectroscopy user supplied): elastic medium transformation glasses, boson peak, elastic medium transformation, Raman light scattering

published or not. The documents may come

Temperature evolution of the boson peak and Debye scaling in vitreous B2O3

Introduction

A common feature of glasses and supercooled liquids is the presence in the vibrational density of states g(ω), of an excess of modes over the Debye level. In Raman and neutron inelastic scattering spectra it appears as a broad maximum in the so called reduced density of states g(ω)/ω 2 , and it is known as boson peak (BP), [1][2][3][4][5][6][7]. Many different approaches have been proposed to explain the origin and the nature of these extra modes, but this topics are still objects of debate, [8][9][10][11][12][13][14][15][16][17][18].

In recent years many experimental efforts have been devoted to understand the evolution of the BP as a function of thermodinamical parameters like temperature [23,24] and pressure [25]. Moreover, the BP has been also studied as a function of density [26][27][28] and time during a chemical vitrification process [29]. A general result is that, when the density is increased, no matter how, the BP position ω BP shifts towards higher frequencies and it decreases its intensity. Moreover, plotting spectra corresponding to different densities as a function of ω/ω BP , they collapse on a master curve [23,27]. This shows that the BP spectral shape is invariant: any intensity variation is only apparent and it is due to the BP shift. The explanation of the origin of this shift is now the open problem. Assuming an acoustic nature for the BP mode [8,9,14], the shift can be explained by considering the elastic medium evolution i.e. the variation of the Debye frequency ω DB [23][24][25][26][27]29]. This approach has been applied to many different systems with contrasting results. As a matter of fact it provides a good scaling only in some cases [24,26,29] can conclude that elastic medium transformation does not always cover the whole story.

In this paper we present a Raman scattering study of the boson peak temperature evolution in vitreous boron oxide v-B 2 O 3 , from the liquid helium up to the supercooled region. Boron oxide is the object of many detailed studies (e.g. references [19][20][21][22], in what follows we point out our attention on two specific points: the existence of the already mentioned master curve and the comparison between BP shift and elastic medium modification.

Experimental details

Vitreous B 2 O 3 is a strong glass with m 32 and a rather low glass transition temperature, T g 576 K [30].

The sample was prepared starting from Aldrich boron oxide powder, purity greater than 99.995%. The powder was dehydrated at 100 • C and then melted at 1100 • C in an alumina crucible. The homogeneous and bubble free melt was subsequently quenched in air. The obtained samples were polished in order to obtain good optical surface.

Raman scattering experiments were performed using a Jobin-Yvon U1000 double monochromator in 90 • geometry. As a light source it was used the 514.5 nm line of an argon ion laser. The spectra were acquired in a wide range of frequencies, from -300 cm -1 to 1300 cm -1 with different resolutions. In the low frequencies region the resolution is about 2 cm -1 . The measurements were performed in both VV and HV polarization geometries, where V and H indicate the polarizations perpendicular and parallel to the scattering plane.

Results and discussion

The first order Raman scattering intensity for a Stokes process can be written as [31]:

I R (ω) = C(ω)g(ω) [n(ω, T ) + 1] ω ; (1)
where n(ω, T ) is the Bose-Einstein population factor and C(ω) is the lightvibrations coupling function. Dividing the Raman intensity for [n(ω, T ) + 1] and ω we obtain the so called reduced Raman intensity I red (ω), which is proportional to the reduced density of vibrational states g(ω)/ω 2 , namely

I red (ω) = C(ω) g(ω) ω 2 ; (2)
Reduced Raman depolarized (HV) spectra are reported in figure 1 for selected temperatures. Spectra at different temperatures are normalized to the same total area in order to be compared. High temperature spectra are characterized by the rise of a quasielastic scattering signal (QES) which slightly modifies also the BP parameters. The QES contribution has then been subtracted using the procedure described in ref. [23]. It can be seen that, when the temperature is increased, hence the density decreases, the BP peak shifts towards lower frequencies and increases its intensity. In the temperature range investigated, the position shift is ∼ 20% while the intensity shift is ∼ 50%. To determine the peak frequency ω BP we fit each spectrum using a log-normal function. We start our analysis of present data by considering the BP spectral shape by using a rescaled frequency ν = ω/ω s and performing the variable transformation g(ν)dν = g(ω)dω. Assuming that, in the BP frequency range C(ω) ∼ ω, [32,33], the squeezed intensity can be written as:

I s (ν) = I red (ω) × ω 2 s . (3)
The so squeezed spectra are reported in figure 2; within the experimental uncertainty they collapse on a single master curve without any adjustable parameter. The frequency ω s is defined as the frequency which allows this scaling for each spectrum. In this case ω s = ω BP , as it is clearly visible in the inset of figure 2. The small differences, smaller than the uncertainty, could be ascribed to the difficulty in obtaining Raman spectra in absolute intensities. The existence of this master curve shows that the BP evolution is ruled by its shift. To understand the origin of this feauture, we compare ω BP to the elastic properties of the systems, which are described trough the Debye frequency ω DB . The Debye frequency has been computed using sound velocities from ref. [34], measured by Brillouin light scattering, and density from [35]. The T evolution of the sound velocities is reported in the inset of figure 3 where a transverse sound wave is still visible also in the supercooled and liquid region, up to ∼ 1300 K. This allows the use of the Debye frequency formula also above the glass transition temperature. In the explored T range the density decreases of about 18%. The comparison beteween ω BP and ω DB is reported in figure 3: it can be seen that they exhibit the same T behavior. The boson peak position scales with the Debye frequency ω BP , i.e. the transformation visible in the reduced density of states g(ω)/ω 2 can be ascribed to the elastic medium modifications.

Conclusions

We investigated the temperature evolution of the boson peak in vitreous B 2 O 3 . We find a master curve for the BP, showing the invariance of the shape. Moreover we find also that the BP shift in this glass could be accounted using the elastic medium transformation approach. Similar studies performed in other strong systems, like vitreous silica [23,28] and vitreous germania [27], pointed out the absence of such a scaling property, thus invocating the existence of further mechanisms that rule the BP shift. This completely different result can add a piece to understand the nature of these processes.

Figure 1 .

 1 Figure 1. Reduced depolarized Raman intensity for the v-B 2 O 3 , symbols are as in the caption; the dashed line is a guide for the eyes and it represents the temperature evolution of the BP.

Figure 2 .

 2 Figure 2. Master curve obtained as described in the text, symbols are as in the caption; in the inset is reported a comparison between ω BP and ωs, respectively indicated with dots and open circles.

Figure 3 .

 3 Figure 3. Comparison between ω BP (dots) and ω DB (open circles); longitudinal (open diamonds) and transverse (open squares) sound velocity are reported in the inset [34].

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Frans R Spiering
email: f.spiering@science.ru.nl

Masha B Kiseleva

Nikolay N Filippov

Bas Van Lieshout

Adriaan M H Van Der Veen

Wim J Van Der Zande

The effect of collisions with nitrogen on absorption by oxygen in the A-band using cavity ring-down spectroscopy

Keywords: Oxygen, A-band, Collision Induced Absorption, Cavity ring down spectroscopy, Line Mixing Oxygen, A-band, Collision Induced Absorption, Cavity Ring Down spectroscopy, Line Mixing

The effect of collisions with nitrogen on absorption by oxygen in the A-band using cavity ring-down spectroscopy.

Satellite measurements are used extensively for climate studies. In particular, a large interest exists in accurate measurements of CO 2 emissions and sinks at the surface of the earth [START_REF] Crisp | The Orbiting Carbon Observatory (OCO) mission[END_REF]. In view of the present column densities, such measurements must have an accuracy of better than 1%. To achieve this level of accuracy, the consequences of clouds reducing the visibility of the surface, and of aerosols creating extra path lengths must be quantified. Both of these effects can be accounted for by making use of spectral windows. The oxygen A-band is such a window. In this spectral region, absorption is due to O 2 molecules only. Furthermore the distribution of oxygen throughout the atmosphere is very well known, allowing for determinations of cloud top heights and aerosol scattering as well as air mass retrieval, by comparing the retrieved surface pressure with the actual pressure. The desired accuracy for CO 2 retrieval requires oxygen measurements with an even better accuracy. For the A-band highly accurate line shapes are needed beyond the simple Voigt line shapes. Effects that affect the line shapes are Dicke narrowing [START_REF] Dicke | The effect of collisions upon the doppler width of spectral lines[END_REF] and speed dependent effects of collision broadening [START_REF] Fowler | Doppler and collision-broadening effects in the profile of spectral lines[END_REF][START_REF] Lovett | Implications of a speed dependent modified voigt function in atomic absorption spectrometry[END_REF]. These effects change line shape near the line centres. The valleys in between the lines are affected mostly by line mixing (LM) [START_REF] Hartmann | Collisional Effects On Molecular Spectra[END_REF] and collision induced absorption (CIA) [START_REF] Frommhold | Collision-induced Absoption in Gasses[END_REF]. In 2005, an analysis of high resolution solar spectra revealed that errors in photon paths up to 5% were found analysing the A-Band in combination with Voigt lines and it was concluded that the description of the absorption far from the line centres needed improvement [START_REF] Yang | Ground-based photon path measurements from solar absorption spectra of the O-2 A-band[END_REF]. Tran and Hartmann showed that a large part of the discrepancy could by explained by the absence of line mixing and collision induced absorption [START_REF] Tran | An improved O 2 A band absorption model and its consequences for retrievals of photon paths and surface pressures[END_REF].

Voigt line shapes result from the convolution between a Gaussian line shape, accounting for molecular motion (Doppler broadening), and a Lorentz line accounting for molecular collisions (pressure broadening). Molecular dependent coefficients for pressure broadening are found in the HITRAN database [START_REF] Rothman | The HITRAN 2008 molecular spectroscopic database[END_REF]. LM describes the consequences of inelastic collisions on a spectrum. Intensity shifts from the wings of the band to the centre of the band and the line shape becomes non-Lorentzian. On average, the intensity is redistributed and the integrated band strength is thus not affected. On the other hand, CIA describes the absorption by transient or metastable collision complexes. The long range quadrupole-quadrupole interaction during the collisions between oxygen and oxygen and nitrogen and oxygen lead to extra transition probabilities. The magnitude of the CIA between oxygen and nitrogen (O 2 N 2 CIA ) seems better understood than the CIA due to oxygen oxygen collisions (O 2 O 2 CIA ) for which no satisfactorily model exists [START_REF] Tipping | Collision-induced absorption in the a1[END_REF][START_REF] Tipping | Theoretical analysis of the collision-induced electronic absorptions in o2-n2 and o2-co2 pairs[END_REF]. As the interaction times during collisions are of the order of picoseconds or less, the CIA spectrum is expected to be broad and without rotational structure. We note, that the CIA strength scales strictly quadratically with collision rate. Its relative impact to the total extinction increases at higher densities, an aspect of which we make use in our analysis.

In recent times the A Band has been studied in considerable detail, mostly with Fourier Transform Spectroscopy (FTS) techniques. The CIA of oxygen with various partners (oxygen, nitrogen, carbon dioxide) has been quantified at high pressures [START_REF] Tran | An improved O 2 A band absorption model and its consequences for retrievals of photon paths and surface pressures[END_REF][START_REF] Tabisz | Interpretation of visible and near-infrared absorption spectra of compressed oxygen as collision-induced electronic transitions[END_REF][START_REF] Tabisz | Pressure dependence of the electronic transition (1)Sigma(+)(g) ¡-(3)Sigma(-)(g) in the absorption spectrum of compressed oxygen[END_REF][START_REF] Tran | Line mixing and collision-induced absorption by oxygen in the A band: Laboratory measurements, model, and tools for atmospheric spectra computations[END_REF][START_REF] Vangvichith | Line-mixing and collision induced absorption for O2-CO2 mixtures in the oxygen A-band region[END_REF]. Tran and Hartmann found that a LM model is essential to retrieve a consistent CIA spectrum. At atmospheric pressures, a highly accurate line shape analysis was performed also indicating the need for LM apart from subtle effects as Dicke narrowing and speed dependent effects [START_REF] Predoi-Cross | Spectroscopic lineshape study of the self-perturbed oxygen A-band[END_REF]. Earlier studies that significantly improved the HITRAN database also used FTS [START_REF] Brown | Experimental line parameters of the oxygen A-band at 760 nm[END_REF][START_REF] Robichaud | High-accuracy transition frequencies for the O 2 A-band[END_REF]. In FTS studies at low pressures (up to several bars), CIA is difficult to retrieve because of base line uncertainties [START_REF] Chagas | Impact of new measurements of oxygen collision-induced absorption on estimates of short-wave atmospheric absorption[END_REF]. In a previous article from our group [START_REF] Spiering | Line mixing and collision induced absorption in the oxygen a-band using cavity ring-down spectroscopy[END_REF], referred to as FS, we confirmed the need for LM and determined the O 2 O 2 CIA at atmospheric relevant densities using cavity ring down spectroscopy (CRDS). Accurate data were taken with a a 0.07 cm -1 bandwidth laser in between the lines of P-branch, the region of the missing Q-branch and frequencies above the band head in the R-branch. The LM model we used was developed by Tonkov and co-workers from the co-authors of the Saint Petersburg group. Its parameters are taken directly from the HITRAN database [START_REF] Rothman | The HITRAN 2008 molecular spectroscopic database[END_REF]. In FS the experimental and theoretical methods are described in detail. In the present article, we expand the work performed in FS by looking at the CIA resulting from O 2 -N 2 collisions in the oxygen A-band studied in three different mixtures.

Theory

The true line shape of allowed transitions is not simple to describe. A Voigt line shape assumes the Doppler effect and the collisional broadening to be independent. Many models have been developed to introduce detailed effects of collisions and the dependence of the speed of the molecules on absorption and thus the line shape. Models of the Dicke effect and of speed dependent effects change the line shapes near the line centre. LM is an effect that has a relatively large effect in between the lines and above the band head of the R branch. We concentrate on the valleys in between the lines and hence we can restrict our model to LM as modification of Voigt type line shapes. The LM model we used is called the adjustable branch coupling (ABC) model [START_REF] Bulanin | Influence of line interference on the vibration-rotation band shapes[END_REF][START_REF] Tonkov | A simple model of the line mixing effect for atmospheric applications: Theoretical background and comparison with experimental profiles[END_REF]. In this model, the collisions are described as hard collisions. The ABC-model has only one free parameter, the so called interbranch coupling parameter (r), all other parameters come from HITRAN. In one extreme no distinction is made between lines in the different branches (r = 1). In the other extreme, line mixing is only allowed within the P-branch and within the R-branch but not in between the branches (r = 0). As expected, this parameter mostly affects the predicted absorption in the missing Q-branch. For an extended description of the model, we refer to FS and the references therein. The effect of foreign gas such as N 2 is assumed to be additive. The collision contribution is weighed by the volume fractions, (V k ). The total broadening for a given line (γ j ) is thus given by

γ j = V O2 γ j,O2 + (1 -V O2 )γ j,N 2 , (1)
where V k is the volume fraction of gas k and γ j,k is the broadening of a particular line j due to gas k. In the HITRAN database [START_REF] Rothman | The HITRAN 2008 molecular spectroscopic database[END_REF][START_REF] L S Rothman | The HITRAN 2004 molecular spectroscopic database[END_REF] the broadening due to O 2 (self broadening) and air are included. From the air broadening, it is possible to retrieve the broadening due to N 2 , when one considers air to consist of 21% O 2 and 79% N 2 .

Fano was the first to derive an expression for the effect of LM on Lorentz lines [START_REF] Fano | Pressure broadening as a prototype of relaxation[END_REF]. He derived the following expression for the spectral distribution:

Φ(ω) = 1 π e    qj,q j ρ j x qj x q j 1 i (ω -L 0 ) + Γ(ω) qj,q j    . (2)
In this equation, ρ j are the (temperature dependent) initial state populations, x qj are the reduced dipole moments. The products ρ j |x qj | 2 provide the line strengths as found in hitran [START_REF] Rothman | The HITRAN 2008 molecular spectroscopic database[END_REF][START_REF] L S Rothman | The HITRAN 2004 molecular spectroscopic database[END_REF], ω is the frequency considered. The labels q and q are used to distinguish transitions within the two individual branches, the remaining label j simply enumerates all optical transitions. L 0 is also known as the Liouville operator, which is a diagonal matrix containing all transition frequencies. The matrix Γ contains all collisional effects and is called the relaxation matrix. The diagonal of the relaxation matrix contains pressure shifts and broadenings and the off-axis elements contain line mixing effects. Hence, a diagonal relaxation matrix describes a spectrum being a simple sum of Lorentzian lines.

The hard collision model implies a direct relation between the collision rate and the LM effect. The elements of the matrix Γ are given by [START_REF] Bulanin | Influence of line interference on the vibration-rotation band shapes[END_REF]:

Γ qjq j = (1 -r)δ qq τ -1 δ jj -A -1 q ρ j x qj x q j + rτ -1 δ qq δ jj -A -1 ρ j x qj x q j +iδ jj ∆ν j , (3)
In this equation τ -1 is the averaged collision frequency, which is obtained from the pressure broadening parameters from hitran, and ∆ν j is the pressure shift of line j. We use these pressure shifts as found in hitran database. A is the total integrated intensity of the band and A q is the integrated intensity of branch q.

All components of the relaxation matrix can be complex, accounting for both line shifts and changes in line shape. We ignore the imaginary part of the diagonal elements, which reflect the pressure induced line shifts, in the calculation of the off-diagonal components. The only free parameter is r, which makes it possible to affect the amount of coupling between the two branches. After using the ABCmodel to introduce line mixing, the Doppler effect is accounted for in a convolution procedure. For the data treatment, the absorption coefficient is calculated for 100 densities in each pressure scan. These model absorption coefficients are subtracted from the measured extinction coefficients to determine the CIA contribution.

Experiment

In order to determine the extinction for O 2 mixed with N 2 , three mixtures were used, two of which were quantified by the Netherlands Institute for Standards (NMI). The mixing ratio's are shown in table 1.

The total extinction due to these mixtures were measured at selected wavelengths using cavity ring down spectroscopy (CRDS). Light from a pulsed nanosecond laser is injected in a cavity with highly reflective mirrors (R ∼ 0.99995) The decay of this light through one of the mirrors is measured using a photomultiplier tube (PMT) selected for its linear response. The decay constant is determined by the reflectivity of the mirrors and the absolute extinction due to the gas mixture in the cavity. By making use of pressure ramps and starting at near zero pressure the mirror reflectivity is explicitly measured at each frequency. The experimental set-up is identical to that used in FS. After having evacuated our pressure cell, we slowly filled it with the gas mixture, while continuously measuring the ringdown time, laser frequency, and gas pressure, all simultaneously. In order to reduce errors in the O 2 N 2 CIA due to possible errors in the O 2 O 2 CIA, measurements on gas mixtures were alternated with measurements on pure O 2 . We found that this procedure gave the same results when using the averaged CIA spectrum of O 2 O 2 deduced in FS.

Results

CRDS determines total extinction coefficients as function of density. The extinction coefficients contain contributions from Rayleigh scattering, from the allowed magnetic dipole absorption of O 2 , from the O 2 O 2 CIA and from the O 2 N 2 CIA . Rayleigh scattering is proportional with density. The CIA contributions are strictly quadratic in density, and the line mixing causes the minima between the lines to behave approximately quadratic in density. In FS we found that the LM model predicts a quadratic dependence in between the lines in the case of pure oxygen. Our LM model predicts that oxygen-nitrogen mixtures do not behave purely quadratic.

Table 2. The refractive index (n) and King correction factor (F )for both O2 and N2 at 288.15 K and 1 atm.

From these the Rayleigh scattering (σ) cross section can be derived using σ = 24π 3 ν4

N 2 n 2 -1 n 2 +2 2
F , where N is the density at 288.15 K and 1 atm. The resulting Rayleigh scattering cross section is independent of temperature and density. To derive the Rayleigh scattering for a gas mixture, the scattering for the individual gases is weighted by their relative volume densities and then added together.

Gas

Refractive Index [START_REF] Bates | Rayleigh-scattering by air[END_REF] King Correction Factor [START_REF] Bates | Rayleigh-scattering by air[END_REF]

(F ) (n -1) × 10 8 O 2 21351.1 + 21.8567×10 12 4.09×10 9 - ν cm -1 2 1.096 + 1.385 × 10 -11 ν cm -1 2 + 1.448 × 10 -23 ν cm -1 4 N 2 6855.2 + 324.3157×10 12 1.44×10 9 ν cm -1 2 1.034 + 3.17 × 10 -12 ν cm -1

2

,The experiments with mixtures are performed to higher densities than our experiments in pure oxygen. Due to the presence of oxygen isotopologues and the use of higher pressures, small deviations from pure quadratic behaviour (order of 0.5-1.5%) were found. Subtraction of the calculated LM values correct for this effect. In FS we have used literature values to subtract the Rayleigh scattering contribution from the total extinction. In this article we retrieve the Rayleigh component from the data. The advantage is twofold. Firstly, the numerical values can be compared with Rayleigh scattering data and secondly, it provides an indication if the data is treated correctly. In experiments with pure dried nitrogen, the accuracy of the retrieved Rayleigh scattering data was found to be of the order of 1 % [START_REF] Spiering | Direct rayleigh scattering measurements on n 2 in the wavelength region 660 -780 nm and on n 2 and o 2 in the 1.2 µm wavelength region[END_REF]. Table 2 summarizes the Rayleigh cross section at the temperature of our experiments (293 K) which were derived from the seminal paper by Bates including the King factor correcting for the anisotropy of the molecules involved [START_REF] Bates | Rayleigh-scattering by air[END_REF].

Determination of the O 2 N 2 CIA and Rayleigh scattering is performed as follows. Firstly, we subtract the allowed magnetic dipole absorption resulting from our ABC model. We are now left with the sum of Rayleigh scattering (linear in density) and the total CIA contribution (quadratic in density) consisting of both the O 2 O 2 CIA and O 2 N 2 CIA contribution. In the next step, we performed a weighted secondorder polynomial fit on these results. In this fit, the zeroth-order results from the mirror offset, the first from the Rayleigh scattering and the second from the (total) CIA. The final step is to subtract the O 2 O 2 CIA contribution, taken from FS, weighted with the appropriate quadratic density. Our consistency check is the magnitude of the derived linear Rayleigh component. The quadratic coefficient is corrected for the the factor D O2 D N2 resulting in the CIA coefficient at a single wavelength. This procedure is repeated at many wavelengths in between the P-lines, in the region of the missing Q-branch and above the R-branch. No measurements were performed in the between the lines of the R-branch. Due to the close proximity of the individual lines in the R-branch (±2 cm -1 between lines), the resulting ring down signals are no longer mono-exponential and, as such, cannot be correctly analysed by our method.

In Fig. 1 the O 2 N 2 CIA for the three mixtures is shown. Both the magnitude and the shape of the curves are the same for all three mixtures, which reveals that the data treatment is consistent. The absolute absorption coefficients are only of the order of 10 -8 cm -1 amg -2 . 1 We note that the O 2 N 2 CIA is only a small fraction of the total observed absorption, which explains partially the statistical spread in the data. The negative values above the bandhead and in the Q-branch do reflect a systematic error, as the O 2 N 2 CIA values should be positive. We have looked at various possible sources for this effect. For the Q-branch, setting the Wavenumber (cm -1 ) Collision Induced Absorption (cm -1 amg -2 ) Figure 1. The O 2 N 2 CIA for the three mixtures (see table 1). Cirlces: mixture one. Triangles: mixture two. Squares: mixture three.

Rayleigh scattering to the theoretical value (table 2) increases the overlap between the measurements as well as negating the negative values. This procedure has no effect on the other measurements. For the measurements just beyond the R-branch bandhead, replacing our earlier O 2 O 2 CIA values by those published by Tran et al. does not mitigate this problem. In principal, we have only one free parameter, the interbranch coupling parameter, r. This parameter also affects the O 2 O 2 CIA and variation showed that the most consistent data were obtained keeping the value fixed to that found in our work on pure oxygen (r=0.8).

We have simulated the effect of a systematic error in the pressure reading. The observed pressure determines how much of the observed absorption is ascribed to the magnetic dipole transition and the O 2 O 2 CIA . An overestimation of the pressure results in O 2 N 2 CIA values that are too small. In the final error analysis, we have allowed for an uncertainty in the pressure of ±1%. It is likely that a refinement of the LM model or modifications of the line parameters may remove the unphysical negative values. We note that all mixtures consistently give near zero O 2 N 2 CIA values below 13000 cm -1 and above 13200 cm -1 , in contrast to the O 2 O 2 CIA spectrum as will be shown below.

Figure 2 shows the average Rayleigh Scattering (RS) retrieved from the observations and compared to the computed values. The error bars reflect the statistical spread. These data may well be the first retrieval of absolute RS values below a magnetic allowed transition. The retrieved RS spectra have a small systematic component which follows the shape of the P-branch. The RS values of mixture two are smaller than expected. Although the RS contribution is retrieved from the linear dependence in density and the CIA contribution is quadratic in density, it is noteworthy that the apparent RS values increase close to the bandhead where the apparent CIA values become negative.

In Fig. 3 we show the averaged O 2 N 2 CIA resulting from the three different gas mixtures. For an estimate of the error bars, we allowed for the possible ± 1% error in the pressure readings. The error indicated is taken to be the largest of the possible error from the pressure readings or the observed statistical error. In order to estimate the integrated CIA band coefficient as well as provide a full O 2 N 2 CIA spectrum, we fitted our observed CIA spectrum with a fit based on the A-band spectrum using Gaussian lines broadened to a width of 7 cm -1 (20 cm -1 FWHM), a procedure suggested by Tabisz et al. [START_REF] Tabisz | Interpretation of visible and near-infrared absorption spectra of compressed oxygen as collision-induced electronic transitions[END_REF] for generating CIA spectra. The resulting fit is displayed in Fig. 3 the P-branch remarkably well. The total integrated absorption including the part under the R-branch yields (1.24 ± 0.3) × 10 -5 amg -2 , which agrees reasonable well with the value obtained with the value of (0.96 ± 0.1) × 10 -5 amg -2 reported by Tran et al. [START_REF] Tran | Line mixing and collision-induced absorption by oxygen in the A band: Laboratory measurements, model, and tools for atmospheric spectra computations[END_REF]. The difference can be understood by comparing our results with the CIA spectrum retrieved by Tran et al. with a 0.5 cm -1 resolution FT instrument at several pressures between 50 and 150 amg (see Fig. 3). Their O 2 N 2 CIA spectrum predicts smaller values below the R-branch ((0.42 ± 0.1) × 10 -5 amg -2 ) than below the P-branch ((0.54 ± 0.1) × 10 -5 amg -2 ). Integrating our spectrum from 129750 cm -1 to 13123 cm -1 yields (0.63 ± 0.3) × 10 -5 amg -2 , reducing the difference between the two experiments. Our spectrum is more structured and predicts a minimum near the Q-branch. The LM-models used by Tran et al. and here are very similar. A small difference is that in Tran et al. a smoothing has been applied to the O 2 O 2 CIA spectrum which may also affect their O 2 N 2 CIA spectrum slightly. In view of the very different experimental methods and the very different pressure ranges of both studies, the agreement in the spectrum and the integrated value is surprisingly good. We stress that to our knowledge our method is the first to determine the CIA below an allowed band using experiments at single frequencies. When comparing the spectrum of O 2 N 2 CIA with the O 2 O 2 CIA (see Fig. 4), large differences can be seen. The largest difference is found above the bandhead. The O 2 O 2 CIA is slowly decreasing, while the O 2 N 2 CIA almost directly goes to zero. The wings are often explained invoking exchange between photon energy and the kinetic energies of the collision partners. Whether excitation energy and kinetic energy can be exchanged, depends on the change in the intermolecular potential during the excitation process and the impact parameter of the collisions involved. As had been concluded already in earlier theoretical treatments of Tipping and co-workers, the mechanism that generate the O 2 N 2 CIA spectrum is insufficient to describe the O 2 O 2 CIA completely. We note here that whereas the average photon energy < ν > of the O 2 O 2 CIA spectrum is larger than the average photon energy of the allowed magnetic dipole transition, our results indicate that for the O 2 N 2 CIA spectrum the average photon energy is equal to the average of the allowed dipole transition. Here, the average photon energy is defined as

< ν >= ∞ 0 α(ν)νdν ∞ 0 α(ν)dν , (4)
where α(ν) is the absorption at wavenumber ν. In comparison, the CIA spectrum reported by Tran et al. implies that the average photon energy is smaller than that of the allowed magnetic dipole spectrum, as the CIA coefficients below the R-branch are smaller than below the P-branch. On average, in the CIA process involving two oxygen molecules part of the photon energy is converted in either internal rotational energy of the other partner or in extra kinetic energy of the two collision partners, whereas this is not he case for O 2 N 2 CIA. Our experiments unravel the various contributions to the total extinction of solar light around the oxygen A-band spectrum for an Earth-like atmosphere containing 79% nitrogen and 21% oxygen. We show the results on a log scale (Fig. 5). As a reference, Figure 5 also shows the relative contributions in the case of pure oxygen. In both cases the absolute contribution of Rayleigh scattering is approximately the same. As the contribution of the oxygen A-band is a factor five smaller in air, the relative contribution of Rayleigh scattering is significantly higher in the case of air. The CIA contribution of collisions with nitrogen is comparable to the CIA contribution due to oxygen due to the larger nitrogen content of air. The integrated impact of CIA at ambient pressures is small. CIA contributes to about 2% of the The blue line displays the absorption due to magnetic dipole absorption, the purple line collision induced absorption and the green line the Rayleigh scattering. The total extinction is given by the black line. Right: The extinction contributions in a mixture of 21% O 2 and 79% N 2 at 1amg and 296K. The blue line displays the absorption due to magnetic dipole absorption, the purple line the collision induced absorption due to O 2 O 2 complexes, the red line the collision induced absorption due to O 2 N 2 complexes (identical to the fit of Fig. 3) and the green line the Rayleigh scattering. The total extinction is given by the black line. For both figures, the O 2 O 2 CIA spectra results from interpolation of the measurements of FS, except for the R-branch region, which is modelled in the same way as the O 2 N 2 CIA. total extinction. The relative contribution is largest in the region above the Rbranch bandhead and the windows in between the lines. We stress that the effects are small in the case of linear absorption. The CIA becomes quantitatively more relevant in cases where the maxima of the resonances are saturated such as in the Earth atmosphere. As Rayleigh scattering is linear and CIA is quadratic in density, the relative contribution is valid for one density, being 1 amagat in this case. It is of interest to note that in the minima in between the lines LM results in general in a small decrease in the predicted absorption whereas CIA increases the absorption, reducing the error in case both effects are neglected. Nevertheless, recently Tran and Hartmann showed that with their values errors in retrieving local atmospheric pressure values based on the total atmospheric column reduced from larger than 1.5 % to less than 0.5% after inclusion of LM and CIA [START_REF] Tran | An improved O 2 A band absorption model and its consequences for retrievals of photon paths and surface pressures[END_REF].

Conclusion

For determination of CO 2 emissions using remote sensing, accurate oxygen spectra are needed especially far away from the line centres. Line mixing and collision induced absorption must be taken into account. In FS we determined the O 2 O 2 CIA spectrum. In this work, we expanded our measurements to the effects of LM and CIA in mixtures of oxygen and nitrogen, in order to determine the O 2 N 2 CIA.

Using cavity ring-down spectroscopy in combination with density ramps, we determined absolute absorption coefficients for mixtures of O 2 and N 2 in the oxygen A-band (760 nm) at room temperature as function of the density. We employed the ABC-model to describe the absorption spectrum due to magnetic dipole absorption with line mixing and Doppler broadening. We used our knowledge of the O 2 O 2 CIA and deduced the O 2 N 2 CIA and, as a control of our analysis, the Rayleigh scattering cross section. The RS determination may well be the first below an allowed band.

The resulting O 2 N 2 CIA spectrum could be fitted well using Gaussian broadened lines of the magnetic dipole transitions, with resonance frequencies and intensities based on the HITRAN08 database. The FWHM width of the lines turned out to be about 20 cm -1 .

Our method shows to be able to retrieve very small absorption coefficients and to unravel the different processes determining the A band spectrum at ambient conditions. We would like to express our thanks to both referees for the care they have taken to read through our manuscript. In reaction to both referees, in particular to referee 2, we have made a serious series of changes to our manuscript that we discuss in some detail below. We not only have made changes to the manuscript but also have used time to perform several integrals based on the remarks of referee 2 to convince ourselves that we believe in our research strategy.

We hope that with these changes we have reacted carefully on the referee comments. Below, we will provide a general discussion for the points indicated by the referees. Sincerely yours,

On behalf of all authors, Frans Spiering The corrections proposed by the first referee are accepted and incorporated in the text: Page 2, line 19 should read.. interaction times… The sentence previously read: "As the interaction time during a collision are..." and now reads: "As the interaction times during a collision are...".

Page 3 line 16. Shouldn't the term times (1-DO2) be gi,N2? The referee is correct and formula (1) is changed to the correct version. The equation is furthermore changed due to remarks of the second referee. These changes will be explained later.

Page 3 line 18 should be left justified. This part of the text has undergone some rewriting, for which an explanation will be given at the corresponding point of referee 2.

Page 4 line 52 should read …do not… The sentence previously read: "Our LM model predicts that oxygen-nitrogen mixtures no not behave purely quadratic." and now reads: "Our LM model predicts that oxygen-nitrogen mixtures do not behave purely quadratic.".

Page 5 line 22/23 should read,,,second-order polynomial…zeroth-order results… The sentence previously read: "weighted second order polynomial fit on these results. In this fit, the zeroth order results..." and now reads: "weighted second-order polynomial fit on these results. In this fit, the zeroth-order results...".

Page 5 line 39 the exponent on amg should be -2 not -21 This error results from the footnote regarding the amagat. In order to make clear that the footnote is not part of the power, the dot to end the sentence has been moved to in between the 1 and 2, i.e. 2 . 1 Page 6 line 44 should read… line parameters… The sentence previously read: "modifications of the line parameters may..." and now reads: "modifications of the line parameters may...".

Based on the comments provided by the second referee, we have made the following changes in the manuscript:

Abstract, last line: I think "to the absorption" at the end of the sentence should be removed The suggestion is followed and "to the absorption" has been removed. Page 1, line 38: Not only the CO2 sources but also the sinks are of interest. I thus suggest "emissions and sinks". I also think adding reference(s) to the OCO or GOSAT satellite projects may be of interest to the reader. The suggestion is followed and "and sinks" is added. Furthermore, a reference to OCO is now included.

Page 1, around line 41: Even in the absence of aerosols and clouds the O2 A band is of interest for the check of the total air mass since it enables a check or retrieval of the surface pressure. The suggestion is followed and "as well as air mass retrieval, by comparing the retrieved surface pressure with the actual pressure." is added to the line.

Page 1, line 53: "Recently, in 2005". Recently is unnecessary here. Recently is removed from the line.

Page 2, line 2:

The authors should mention that it was demonstrated in Ref. [START_REF] Tabisz | Pressure dependence of the electronic transition (1)Sigma(+)(g) ¡-(3)Sigma(-)(g) in the absorption spectrum of compressed oxygen[END_REF], that the errors in te photon path obtained in ref. [START_REF] Frommhold | Collision-induced Absoption in Gasses[END_REF] were due to the use of Voigt shapes and the neglect of both LM and CIA. The referee is correct and the following sentence is added: "Tran and Hartmann showed that a large part of the discrepancy could by explained by the absence of line mixing and collision induced absorption [START_REF] Tran | An improved O 2 A band absorption model and its consequences for retrievals of photon paths and surface pressures[END_REF]." A reference is added as well to support the statement. the N2 value. The sentence read: "In the HITRAN database [START_REF] Yang | Ground-based photon path measurements from solar absorption spectra of the O-2 A-band[END_REF][START_REF] Bulanin | Influence of line interference on the vibration-rotation band shapes[END_REF] both broadening due to O2 (self broadening) and N2 are included." and now reads: "In the HITRAN database [START_REF] Rothman | The HITRAN 2008 molecular spectroscopic database[END_REF][START_REF] L S Rothman | The HITRAN 2004 molecular spectroscopic database[END_REF] the broadening due to O2 (self broadening) and air are included. From the air broadening, it is possible to retrieve the broadening due to N2, when one considers air to consist of 21% O2 and 79% N2." Page 3, line 30: In this equation, the indexes used for the matrices are qj,q'j' whereas qq',jj' is used in Eq. ( 3). This should be corrected for consistency. I think the first one is more correct. We agree with the referee and updated the equation according to the suggestion. Page 3, line 39: "collisional effects" would be more appropriate than "collision dynamics" "Collisional dynamics" is replaced by "collisional effects." Page 3, line 41: "coefficients" is not correct. In fact "broadening coefficient" is usually used to designate the density (or pressure normalized values (eg in cm-1/atm). Since the relaxation matrix in Eq. ( 2) contains the density it would be better to say "collisional shift and width". Note that saying somewhere that the relaxation matrix is proportional to density my be considered. We thank the referee for pointing this out and updated the sentence. It now reads: "...contains pressure shifts and broadenings and..." where it previously read: "...contains pressure shifts and collisional broadening coefficients and...". Page 3, line 45: adding a reference (from the St Petersburg group) where Eq. ( 3) was first introduced should be considered. A reference is now included at the end of the sentence.

Page 3, line 57: I think it would be better to say that the enture imaginary part of the relaxation matrix is neglected. In fact, since it is said, just above, that the collision frequency tau is taken from the broadening, it is obvious that the relaxation matrix in Eq. ( 3) is purely real.. We changed the formula to include the pressure shifts as well, since they are also used. These complex numbers are only located on the diagonal. Additional lines are included to describe the symbol used for the pressure shift in formula [START_REF] Fowler | Doppler and collision-broadening effects in the profile of spectral lines[END_REF]. The additional lines are: "and Deltaj is the pressure shift of line j. We use these pressure shifts as found in hitran database" Page 3, line 59: I have not found, in the paper, any place explaining how the value of r is determined and what value is obtained. This should be explained and the value of r should be given (note that comparing it to the value obtained for pure O2 may be of interest). My guess is that r is chosen to make the CIA as smooth as possible but I am not sure. The referee is correct and we thank him for pointing this out. We replaced on page six the last sentences of the second paragraph. Previously it read: "The only free parameter in our LM-model, the interbranch coupling parameter, r, affects both the O2O2 CIA and the O2N2 CIA values. The systematic negative values cannot be altered without resulting in an unphysical O2O2 CIA spectrum." and it now reads: "In principal, we have only one free parameter, the interbranch coupling parameter, r. This parameter also affects the O2O2 CIA and variation showed that the most consistent data were obtained keeping the value fixed to that found in our work on pure oxygen (r=0.8).". Page 4, line 52: Here I think there may be a serious problem. First, let me say that I guess the correct sentence is "do not behave" and not "no not behave". If this is right, I see absolutely no reason why the LM<model would not predict a quadratic dependence for N2 while it was the case for O2. in fact this quadratic dependence comes from the fact that absorption in the troughs between lines is considered at distances from line centers significantly greater than the widths. As long as this is the case, a quadratic dependence is obtained with a LM model, regardless of the relaxation matrix. Furthermore, the relaxation matrices for collisions with O2 and N2 should be quite alike. But maybe the sentence does not apply to the LM absorption but to the total absorption, including the Rayleigh… This really needs to be clarified. The referee was surprised about a non quadratic nature of our extinction in the case of O2-N2 mixtures. He/she is correct that we should not suggest a fundamental difference, therefore we rephrased this part as: "The experiments with mixtures are performed to higher densities than our experiments in pure oxygen. Due to the presence of oxygen isotopologues and the use of higher pressures, small deviations from pure quadratic behaviour (order of 0.5-1.5%) were found. The authors discuss here the effect of an error on pressure on the retrieve CIA and find that the consequences are negligible as can be expected from the accuracy of the pressure measurement. I am convinced that, as stated by the authors; the CIA is mostly sensitive to the description of the contribution (affected by LM) of the allowed lines. For instance, a change of 2% of the line intensities (and similarly of the half width) will change the allowed absorption in the troughs by 2% and then affect the retrieved CIA. Furthermore, any error in the LM model will have consequences on the CIA. The negative values obtained likely mostly result from these uncertainties, which have emphasized effects on the CIA since the allowed contribution is dominant. This should be clearly stated. In fact, the authors may consider comparing the CIA retrieved after changes of the intensities and/or half widths, and or value of the r parameter. We agree with the referee that the line parameters are important on the influence of the CIA spectrum. We also agree that the LM model probably needs some improvements, but we have already mentioned that in our previous paper. Finally, we decided only to mention the influence of the pressure, since this is a possible source of errors. Page 6, line 52: The residual shape in the Rayleigh probably comes from uncertainties on the measurements. In fact, it is well known that, if a finite number of densities is used, there are correlations between the liner and quadratic coefficients obtained from a polynomial fit. A small change of the values for high densities, for instance, will translate in all three parameters of the second order polynomial. In fact, the residual "P branch shape" in Fig. 2 is only obvious for the circles and, furthermore, much smaller that the error bars. The authors this should not put to much emphasis on this discussion. We thank the referee for pointing this out and we are familiar with mentioned effect, but we decided not to change the text.

Page 7, Figure 3 and discussion starting on line 24: Using the allowed band to compute the shape of the CIA is extremely surprising to me. In fact, the calculated shape, that clearly shows two distinct branches, does not look like any CIA observed previously for CIA bands of O2, N2, CO2, … I am surprised that this procedure could have been suggested by Tabisz. The case of the pure O2 CIA in the A band clearly demonstrates that this is not correct. I thus think that this part of the paper should be removed and that a comparisons with the total band intensity of Tran et al is impossible. Nevertheless, one comparison can be made: it consists in calculating the area below the CIA in the spectral region common to both the present paper and Tran et al. The authors could thus integrate their CIA in the intervals where they have measured values, ie [12950,13125] + [13175,13275] and do the same thing with Tran's data and compare the values. From a look at Fig. 3 my guess is that these two values will be quite consistent The method described by Tabisz is given at: Tabisz et al. Canadian Journal of Physics, 1969; 2859-2870. On page 2867 a description of the method identical to ours is given to fit the A-band CIA spectrum. Furthermore, we disagree with the referee concerning the application of the model. The O2 CIA is different from the N2 and CO2 shapes, indicative that a second process is occuring in pure O2. It is not clear which process this is, although suggestions are put forward with regard to the short range interactions. This process seems absent for CIA resulting from O2N2 and O2CO2 collisions. However, we did follow the suggestion to include a comparison based on part of the spectrum. As such we incorporated values for the P-and R-branch for the CIA spectrum obtained by Tran et al and compared the value obtained for the P-branch with our spectrum integrated over the P-branch. This is expressed in the manuscript in the following way: "Their O2N2 CIA spectrum predicts smaller values below the R-branch ((0.42±0.1)×10-5amg-2) than below the P-branch ((0.54 ± 0.1) × 10-5amg-2). Integrating our spectrum from 129750 cm-1 to 13123 cm-1 yields (0.63 ± 0.3) × 10-5amg-2, reducing the difference between the two experiments." Finally, we noted that the value obtained for our full integrated spectrum was still in atmosphere instead of amagat, which we now replaced by the correct value. The referee is correct in noticing that this discussion is absent in the paper of Tran. We changed the sentence to: "In comparison, the CIA spectrum reported by Tran et al. implies that the average photon energy is smaller than that of the allowed magnetic dipole spectrum, as the CIA coefficients below the R-branch are smaller than below the P-branch." Page 9 discussion starting on line 43: See remark above We hope our explanation above for Page 7, Figure 3 and discussion starting on line 24 is sufficient to support this conclusion.

1 )Figure 2 . 2 )Figure 3 .

 1223 Figure 2. The observed averaged RS (cross) and theoretical value (solid line) for pure O2, mixture one (circles) and (dotted line), mixture two (triangle) and (dashed line) and mixture three (square) and (dotdashed line) .

2 )Figure 4 .

 24 Figure 4. The averaged O 2 N 2 CIA (cirlce) and O 2 O 2 CIA (cross) spectra at 293 K. The O 2 O 2 CIA spectrum is taken from FS[20].

Figure 5 .

 5 Figure 5. Left: The extinction contributions in pure O 2 at 1amg and 296K.The blue line displays the absorption due to magnetic dipole absorption, the purple line collision induced absorption and the green line the Rayleigh scattering. The total extinction is given by the black line. Right: The extinction contributions in a mixture of 21% O 2 and 79% N 2 at 1amg and 296K. The blue line displays the absorption due to magnetic dipole absorption, the purple line the collision induced absorption due to O 2 O 2 complexes, the red line the collision induced absorption due to O 2 N 2 complexes (identical to the fit of Fig.3) and the green line the Rayleigh scattering. The total extinction is given by the black line. For both figures, the O 2 O 2 CIA spectra results from interpolation of the measurements of FS, except for the R-branch region, which is modelled in the same way as the O 2 N 2 CIA.

Page 2 , Line 6 :

 26 The reason why different molecules have different broadenings is not only due to the different masses. It also results from the rotational structure. The sentence is removed.Page 2, line 8: Note that, for a given active molecule, HITRAN only provides the selfbroadening coefficient at room temperature and the value (plus the T dependence coefficient) for air

  Subtraction of the calculated LM values correct for this effect. " Page 5, line 54:

Page 17 of 17 URL

 17 : http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics

  the validity of this discussion but I found it a little "strange". The opinion of specialists in CIA (R.H. Tipping ?) my be of interest. One thing: my memory may be wrong but I do not remember any such discussion in Tran's paper (see Line 30 on page 8).

	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

Page 7, discussion starting on line 46: My knowledge on CIA is not sufficient to evaluate

The dimensionless unit amagat (amg) is defined as the ratio of the measured density to the density of the standard atmosphere.

URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics

We thank Prof. Dr. Tonkov of the Saint Petersburg University for his inspiration. We thank Dr. Tran and Dr. Hartmann for providing us with numerical data of their CIA spectra of pure O 2 . We acknowledge the financial support from NWO (Echo project).

broadening (assumed 21% O2 and 79% N2). This point is discussed at the remark concerning page 3, line 20.

Page 2, line 21: The sentence may be misleading. In fact the CIA increases the total extinction just because CIA exists, and not because it is quadratic in density (if it were linear in density it would increase the extinction as well). The sentence is change from"...and hence with the density and increases the total extinction, an aspect of which we make use in our analysis." to "Its relative impact to the total extinction increases at higher densities, an aspect of which we make use in our analysis." Page 2, line 32: As written, the sentence is to restrictive and not correct. In fact, this sentence is only correct for FTS measurements under conditions where the CIA contribution is small and comparable to the base line uncertainty. This clearly depends on the density range. In fact, there have been numerous FTS measurements of CIA absorption (eg, the Tran et al for the O2 A band, and other studies, for instance by Lafferty et al on the O2 and N2 CIA fundamental bands) The referee is correct. As such, we added a pressure requirement which is more or less needed to detect the studied effect. The sentence read: "In FTS studies CIA..." and now reads: "In FTS studies at low pressures (up to several bars), CIA..." Page 2, line 39: the sentence is complicated; I suggest something like 'The LM model was developed by the co-authors from the St Petersburg group. The sentence is indeed complicated, so we have changed the sentence to be clearer from:"The LM model we used was developed by Tonkov and co-workers from the group of two of us (NNF and MBK)." to "The LM model was developed by the co-authors from the St Petersburg group.", in accordance with the suggestion. Page 3, line 3: I suggest "coupling parameter r" at the end of the sentence. The suggestion is implemented. The sentence previously read: "...so called interbranch coupling parameter, all..." and now reads: "...so called interbranch coupling parameter (r), all...". Page 3, line 4: For r=0 it is indeed obvious, from Eq. ( 2) that the P and R branches are completely uncoupled. Nevertheless, for r=1 there are some non zero relaxation matrix elements within the P branch and within the R branch. Unless there is a mistake in the equation, the statement "the branches are coupled completely" is incorrect. In fact, the reader would expect for completely coupled branches that only inter-branch mixings would be non zero. Obviously, the meaning of the sentence is not completely clear, so we changed it from: "In one extreme the (P-and R-) branches are coupled completely (r = 1)." to : "In one extreme no distinction is made between lines in the different branches (r = 1)." Page 3, line 8: the value of r not only affects absorption in the trough between P and R branches. It affects all regions. This is correct. However, the influence of r is most obvious in the trough between the P and R branch. The sentence is changed to reflect this behavior. The original sentence read: "...parameter affects the predicted". The new sentence reads: "...parameter mostly affects the predicted..."

Page 3, line 16:

The last gama should be gama_i,N2. Also, the use of D for mole fractions (ie with values between 0 and 1) may be confusing since one expects D to be used for densities. Furthermore, the index i is used in this equation to designate a line while j is used for the collision partner. These two indexes are then used in Eq. ( 2) to designate lines. This is confusing. The error concerning the gamma is discussed in the section regarding the first referee. Secondly, the D has been replaced by a V, which is hopefully clearer in meaning. Lastly, the indexes are changed: j is replaced by k, and i is replaced by j in order to be consistent with the other equations.

Page 3, line 19: I suggest "line i due to gas j" The suggestion is followed and updated with regard to the remarks for page 3, line 16. The old sentence read: "particular line due to gas j." and now reads: "...particular line j due to gas k.". Page 3, line 20: HITRAN does not directly provide the broadening by N2 but the broadening by air. Of course, for O2, since the self broadening is also provided, one can deduced the N2 value. We thank the referee for pointing this out. We checked that the parameters are sufficiently equal that it has no effect on our analysis. Furthermore, a description is included of the procedure to deduce