M Mrad

A Tarhini

Pierre-Yves Bourgeois

Vincent Giordano

PHYSIC OF THE SAPPHIRE WHISPERING-GALLERY-MODE SOLID-STATE MASER OSCILLATOR

The Cryogenic Sapphire Oscillator (CSO) is currently the best available technology that can provide a relative frequency stability better than 10 -15 for integration times between 1 s and 10,000 s. But, the CSO remains a complex instrument requiring multiple loop controls to achieve the best frequency stability. The possibility to use of the sapphire resonator in a self-sustained MASER oscillator constitutes an elegant alternative to the CSO. Here, the sustaining amplification is achieved through the interaction between a high-Q factor whispering gallery mode and the paramagnetic Fe 3+ ions, which are present in small concentration in the sapphire crystal. The Fe 3+ ion exhibits three energy states enabling to realize a self-sustaining solid-state maser. Although, this principle has been already experimentally demonstrated few years ago, its development as a truly usable ultra-stable source has not yet been completed, mainly due to the lack of control of the complex physical phenomena involved. This paper completes the previous theoretical work based on the rate equations model. Here we derive the full quantum equations describing the evolution of the Fe 3+ ions inside the sapphire lattice and submitted to a pump and a maser signal. The influence of the ions concentration and spin-spin relaxation time will be pointed out.

INTRODUCTION

The need for microwave sources exhibiting ultra-high spectral purity and high temporal stability is a key point highlighted by the performance limitations in major strategic applications such as radar, spatial communications, ultra-high accuracy navigation systems, and high bit rate optical telecommunications. The Cryogenic Sapphire Oscillator (CSO) is currently the only available technology able to provide a relative frequency stability better than 1×10 -15 for integration time between 1 s and 10 4 s [1,2,3,4,5,6]. The CSO is a classical transmission oscillator that uses as a frequency determining element a passive cryogenic high-Q sapphire resonator based on a cylindrical dielectric resonator made in a high purity sapphire monocrystal. To provide high Q-factor, the sapphire resonator is excited on a high order electromagnetic mode (Whispering Gallery mode or WG mode) and cooled down near the liquid helium temperature [7]. At 10 GHz, the Q-factor typically reaches 10 9 . Nevertheless, the CSO remains a complex instrument. For technical raisons, mainly to limit the required cooling power, the sustaining amplifier is placed outside the cryostat at room temperature. Two long microwave lines join the cryogenic resonator and the sustaining stage. Any variation in the phase or power of the signal injected in the resonator will produce an oscillator frequency variation limiting drastically the achievable frequency stability. To solve this issues, additional sophisticated electronical controls are mandatory to stabilize both the signal phase and power.

In 1959, Bogle and Symmons [8] discussed in detail the study of the resonance of Fe 3+ in alumina and presented accurate measurements of transition frequencies. We reported in [START_REF] Bourgeois | Proc. 19th EFTF Besanc ¸on[END_REF][START_REF] Bourgeois | [END_REF] the possibility to use the sapphire resonator in another way. Here, the sustaining amplification is achieved through the interaction between a high-Q factor whispering gallery mode and the paramagnetic Fe 3+ ions, which are present in small concentration in the sapphire crystal. The Fe 3+ ion exhibits three energy states enabling to realize a self-sustaining solid-state maser. The population inversion is obtained by submitting the Fe 3+ ions to a 31 GHz signal. The maser oscillation takes place at 12.04 GHz, i.e. the ground state electron spin resonance (ESR) frequency of Fe 3+ . As the sapphire resonator whispering gallery mode exhibits at low temperature extremely low losses, the Fe 3+ ions concentration required to get enough gain is very low : typically, below 1ppm (part-per-million).

Since the experimental demonstration of this new type of solid-state maser [11,12,[START_REF] Mrad | IEEE-EFTF conference[END_REF][START_REF] Didier | [END_REF], few theoretical works have been published to explain its main features [15,16,17]. Particularly, the effectiveness of the maser process, i.e. the output power as a function of the pumping power and the resonator coupling has been derived by using the classical rate equations model [START_REF] Siegman | Microwave Solid State Masers[END_REF] as stated by Siegman in the early 60s.

In this paper, we derive the exact equations describing the sapphire maser operation using a quantum mechanics approach considering the coherences that are created between the different quantum states of the paramagnetic ion. These coherences were assumed negligible in the previous approach based on the rate equations. The effect on the maser operation of these coherences increases as a function of the spin-spin relaxation time and of dopant concentration. Conversely, it decreases as a function of the spin-lattice relaxation time.

THE WG MODE MASER OSCILLATOR

Fe 3+ ions are present in high quality sapphire Hemex crystals in quantities of few parts per million. The Fe 3+ ion in the sapphire lattice presents 3 energy levels at zero-magnetic-field [START_REF] Kornienko | [END_REF]20] corresponding to the classical Bloembergen 3 levels scheme [21] represented in Figure 1. To create the inversion population between the first two levels, Fe 3+ ions are pumped to the third level by applying a microwave signal 31.34 GHz. Then relaxation occurs between E 3 (i.e. state |5/2 >) and E 2 (i.e. state |3/2 >) causing population inversion between states |1/2 > and |3/2 >. The system is able to amplify signal at the Fe 3+ EPR frequency, i.e. 12.04 GHz. If the amplifier gain is sufficient to compensate for the resonator loss, the system becomes self sustained and a stable 12.04 GHz can be extracted from the cryogenic oscillator.

The Whispering Gallery Mode Maser prototype consists in a cylindrical sapphire resonator, 50 mm diameter and 30 mm thick. It is placed in the center of a cylindrical cavity and this assembly is cooled down the liquid helium temperature. The resonator W GH 17,0,0 mode frequency coincides with the Fe 3+ ESR, i.e. 12.04 GHz. Around the pump signal frequency, i.e. 31.3 GHz, the resonator mode spectrum is dense and several high-Q factor modes liying in the ESR bandwidth can be used to pump the ions and get oscillation [22,23]. Depending on the coupling factors, the chosen pump mode and the active ions concentration, the maser signal power detected outside the cryostat can be as high as -30 dBm.

FE 3+ IONS PARAMETERS IN THE SAPPHIRE CRYSTAL

The maser operation depends on the concentration N and relaxation times of the Fe 3+ paramagnetic ions. In our experiments, the sapphire resonator are machined in high quality crystals HEMEX type [24] containing very small amount of defect and paramagnetic impurities. This situation greatly differs from those of previous works undertaken in the 60's for which crystal quality was poor compared to the current technology.

N effective ions concentration :

Quantitative ionic impurities concentration measurements have been conducted on HEMEX crystals with different methods [25]. All these measurements agree within the resolution measurements and give a total iron ion concentration at a level of 2 ppm. The given concentration corresponds to the total amount of iron ions including both Fe 2+ and Fe 3+ , among which only the latter is paramagnetic and gives rise to the maser effect. The effective concentration of Fe 3+ ions has been evaluated in several resonators by measuring the real part of the magnetic susceptibility [26]. It appears the number of ions that effectively contribute, is very small, i.e. N is typically in the range 10-20 parts-per-billion (10-20 ppb). This low value is compatible with the maser signal power experimentally observed [24]. It indicates that in typical HEMEX crystals, Fe 2+ dominates. This was confirmed in 2000 [27] : a conversion Fe 2+ to Fe 3+ was achieved by thermally annealing a HEMEX resonator [28] in air resulting in a large increase of the effective ion concentration and consequently of the maser signal power [26]. This led to assume that the Fe 3+ concentration that can be found in high quality sapphire crystal should be in the range :

10 ppb < N < 300 ppb (1)
3.2. τ 1 longitudinal or spin-lattice relaxation time :

τ 1 describing the time required for the populations to recover their thermal equilibrium after any perturbation. The spinlattice relaxation time for Fe 3+ in sapphire has been measured by several groups [8,24,29] and at the liquid helium temperature is in the range :

5 ms < τ 1 < 20 ms (2)
3.3. τ 2 transversal or spin-spin relaxation time :

τ 2 is the rate at which, the transverse component of the magnetization, exponentially decays towards zero. τ 2 is phenomenologically introduced in the rate equation model as related to the ESR linewidth by : ∆ν = 1/πτ 2 . The transversal relaxation time is ultimately limited by the interaction between neighboring spins : spin-to-spin coupling. According to the Van Vleck's theory [30], the ESR linewidth resulting from the dipole coupling should be proportional to the ions concentration N :

∆ν ≈ 6g 2 β 2 µ 0 N S(S + 1)/h ( 3
)
where g is Landé g-factor, β the Bohr magneton, µ 0 permeability constant and the spin S = 5/2. Applied to the Fe 3+ ions, the equation (3) was only approximately verified for relatively high concentrations [8,20]. For high quality sapphire crystals observed ∆ν are always several tens of MHz wide, and thus can not be attributed to the spin-to-spin coupling nor to the crystal defects, but to the hyperfine interaction between Fe 3+ ion and Al nuclei that surround it.

If the crystal is perfectly homogeneous and presents homogeneous distribution of ions, then the lineshape is determined by the relaxation times and therefore lorentzian lineshapes are a common result. The EPR spectrum is the sum of a large number of lines each having the same Larmor frequency and linewidth. In general, the ions are not all in the same environment, the lineshape is determined by unresolved couplings because the EPR spectrum is the sum of a large number of narrower individual homogeneously broadened lines that are each shifted in frequency with respect to each other [20,24,26].

1ns < τ 2 < 20ns (4)

MASTER EQUATION FOR FE 3+

ρ(t) is the density matrix operator. The diagonal elements ρ ii or n i are the energy level populations, which at thermal equilibrium reduces to N i with :

N j N i = exp(- hν i j k B T ) = e -∆ i j and 3 ∑ i=1 N i = 1 (5)
where ν i j is the Bohr frequency. We will deal with the varying population differences : n i j =ρ ii -ρ j j .

The off-diagonal elements ρ i j = < i|ρ| j >, (i = j) represent the coherence between levels |i > and | j >. We assume solutions of the form : where δ i j are complex slow time-varying functions. The master Liouville-Bloch equations of the system are [START_REF] Godone | Coherent Population Trapping Maser[END_REF] :

ρ 12 =
dn 1 dt = - n 12 -N 12 2τ (12) 1 - n 13 -N 13 2τ (13) 1
- and τ

Ω Rp Im[δ 13 ] -Ω Rs Im[δ 12 ] dn 2 dt = n 12 -N 12 2τ (
(i j) 2
are respectively the spin-lattice relaxation time and the spin-spin relaxation time of the pair of levels (i, j).

Pace [START_REF] Pace | [END_REF] introduced measures showing that relaxation times are of the same order of magnitude. We therefore assume that τ

(12) 1 =τ (23) 1 =τ (13)
1 =τ 1 ∀ i and j. Similarly for the spin-spin relaxation time, we therefore assume that, τ

2 =τ

(23) 2 =τ (13)
2 =τ 2 ∀ i and j. The resolution of the previous system of equations is complex, Therefore we'll focus only the steady state of the system :

dn i /dt = 0 and dδ i j /dt = 0 (7)
In addition, the equations are not independent of each other. A seventh equation is necessary. It corresponds to the sum of population differences such as :

n 12 + n 23 -n 13 = 0 (8)
The relationship [24,[START_REF] Godone | Coherent Population Trapping Maser[END_REF] between Ω 2 Rp and W P is Ω 2 Rp = 2W P /τ 2 , where W P is the absorption probability (same relationship between Ω 2

Rs and W S with that W S is the emission probability).

The stationary solutions of the general model are the following :

n 12 ac = a(3 + 2τ 1 W P )N 12 -2b S τ 1 W P N 23 6 + c + d + 4τ 1 (2τ 1 -τ 2 )W P W S (9a)
n 13 ac = a(3 + 2τ 1 W S )N 13 + 2b P τ 1 W S N 23 6 + c + d + 4τ 1 (2τ 1 -τ 2 )W P W S (9b)
where :

         a = 2 + τ 2 (W S +W P ) b S = 2 + τ 2 (W P -2W S ) b P = 2 + τ 2 (W S -2W P ) c = (8τ 1 + 3τ 2 )(W P +W S ) d = 4τ 1 τ 2 (W 2 P +W 2 S )
To derive the rate equations model (Without coherence) from the general model, we eliminate the terms of coherence, assuming that the coherence time τ 2 tends to zero. The stationary solutions of the rate equations model are :

n 12 sc = 2τ 1 (N 12 -N 23 )W P + 3N 12 4τ 1 (1 + τ 1 W S )W P + 4τ 1 W S + 3 (10a) n 13 sc = 2τ 1 (N 13 + N 23 )W S + 3N 13 4τ 1 (1 + τ 1 W S )W P + 4τ 1 W S + 3 (10b)

THE POWERS

Evolution of the emitted power as a function of pump power

In this paragraph we will defined the relation at the resonator input between the pump power (P p ) and the maser power (P s ). The power that can be emitted by a maser system is define as [START_REF] Siegman | Microwave Solid State Masers[END_REF] :

P s = hν 12 W S n 21 V e f f 12 (11)
V e f f 12 is the effective volume of the whispering gallery mode involved in the emmission process [15]. The emission probability is represented as follow [START_REF] Siegman | Microwave Solid State Masers[END_REF] :

W S = 1 4 γ 2 g signal (ν)Q 0 S H 2 acsignal σ 2 12 ( 12
)
From the definition of the Poynting vector, the power in the resonator is directly proportional to the magnetic field. It leads to :

H 2 acsignal = 2P s Z c S e f f 12 (13)
Therefore :

W S = γ 2 τ (12) 2 Q 0 S σ 2 12 Z c S e f f 12 P s = w S 0 P s (14)
This representation can also be applied to the absorption probability :

W P = γ 2 τ (13) 2 Q 0 P σ 2 13
Z c S e f f 13 P p = w P 0 P p (15) γ is the gyromagnetic factor. S e f f 12 and S e f f 13 are respectively the effective surface of the whispering gallery mode in the meridian resonator plan for the maser signal and the maser pump. Z c is the characteristic resonator impedance, σ 2 i j is the transition parameter between the two levels i and j, which their value can be obtained through the calculation of the spin Hamiltonian. Q 0 S and Q 0 P are the quality factors of the whispering gallery modes for the maser signal (the ions from the second to the first level) and the maser pump (the ions from the first to the third level) respectively [15].

To remain in the validity field, we neglect the second order term and suppose that τ 2 << τ 1 and write C = hν 12 NV e f f 12 w S 0 . From equations (9a, 11, 14 and 15) we can now determined the maser power (P s ) as a function of the pump power (P p ) :

P s = 1 w S 0 (-1+ 1 2 C(N 23 -N 12 ))P p -3 4τ 1 w P 0 (1+CN 12 ) 1 w P 0 +τ 1 (1+x)P p (16) x = 0 Without Coherence Cτ 2 4τ 1 (N 13 + N 23 ) With Coherence
For the production of a maser, we need to calculate the threshold pump power and maximum power of the maser in the resonator.

Ppump

threshold Pump power

The pump power threshold P p T h is when the population inversion occurs, that is to say, when n 12 =0. The rate absorption threshold (W P T h ) can be written,

W P T h = 3 2τ 1 N 12 N 23 -N 12 ( 17
)
Then, substituting the expressions of W P T h (equation ( 17)) and n 13 T h (equation (9b) or (10b)) in the expression for the power pump :

P p T h = 3 2τ 1 hν 13 N 12 NV e f f 13 (18)

Maximum output power

If the power applied to pump ions continues to increase, the system will eventually become saturated. In another form, the power P s in a resonator will be maximum when the pump has a power strong enough. Using the equation ( 16), the maximum power of the signal maser reads :

P s max = lim P p →∞ P s = C 2τ 1 w S 0 N 23 -N 12 1 + x (19)

Outside the resonator

The injected power (P pump ) is recycled by a high-order whispering gallery mode with a coupling coefficient β pump . For [15], the transmitted pump power can be expressed as [START_REF] Siegman | Microwave Solid State Masers[END_REF] :

P p = (1 -( 1 -β pump 1 + β pump ) 2 )P pump (20)
The output power due to stimulated emission of the ions will be recycled within the WG 17,0,0 mode centered on the Fe 3+ EPR due to the high Q-factor, a schematic is shown in figure 2. The output power (P out ) is a function of the different losses and gain of the three-level spin system and also of the whispering gallery mode characteristics. For a maser oscillator this output power can be written as a function of the output coupling β out :

P out = β out β out + 1 P s max (21)
with β out the coupling coefficient at the output of the maser.

RESULTS AND DISCUSSION

To determine an order of these powers, we will consider the parameters at 4.2K of a resonator. The Bohr frequencies are ν 12 = 12.04GHz and ν 13 = 31.32GHz. At thermal equilibrium we have : N 12 = 0.050, N 23 = 0.067 and N 13 = 0.117. The values of the transition parameters σ 2 i j for Fe 3+ paramagnetic ion in the sapphire are σ 2 12 = 2 and σ 2 13 = 0.0026 [8]. The total volume of cristal is equal V = 58.9cm 3 . According [33], the effectif volumes of the WGM are V e f f 12 ≈ 2V 7 and V e f f 13 ≈ 2V 17 . The quality factors are equal Q 0 S = 6×10 8 , Q 0 P = 10 8 , the value of the characteristic resonator impedance is Z c =119Ω. To understand the influence of the ions concentration and coherence time on our models, we plot the maximum emitted power as a function of concentration for three values of τ 2 . The results are given in the Figure 3. The coherence time τ 2 is limited by the interaction between neighboring spins (spin-spin coupling). When the return to the equilibrium of the transverse magnetization component is slow, i.e. long coherence time, the power that can be extracted from the maser is lower than in absence of spin-to-spin coupling.

In order to verify the preceding expectations, we have tested three different resonators with different values of the parameters N, τ 1 , τ 2 . The resonator is loaded into a copper cavity and then cooled to the liquid helium temperature in a cryostat. The maser output power is measured with a microwave power spectrum analyzer while the injected power is measured with a power meter. The characteristics of the three resonators are, Pinocchio (16ppb, 18 ms, 2 ns) [15,24], Jeanne (11 ppb, 18 ms, 2 ns) [15,24], and a particular resonator dubbed Geppetto (150 ppb, 7 ms, 5.9 ns) [26]. In figure 4, we plot the threshold Pump power for the first and the third resonator from equation 20.

In figure 5 we plot the theoretical results (with and without coherence) and the experimental data from equation 21.

Experimentally, the cryogenic resonator is connected to room temperature via a microwave K-cable with 5 dB loss. The resonator is loaded into a copper cavity and then cooled to the liquid helium temperature in a cryostat. Two microwave cables enable us to inject the pump power in the cryogenic sapphire resonator (5 dB loss) and to extract the maser output power (the data of the first and second resonator are 2 dB loss). Therefore, these losses have been deducted from the theoretical curve to consider the microwave cable losses.

APPLICATION TO OTHER IONS

In the case of Fe 3+ , the effect of coherence is limited. Other paramagnetic ions, as Mo 3+ and Cr 3+ , can validate the difference between the two models, using a static magnetic field.

Cavity masers using ruby were developed at the Jet Propulsion Laboratory (JPL) at 960 megahertz (MHz) and at 2388 MHz. The 2388-MHz maser was used to receive and amplify microwave radar echoes from the planet Venus [START_REF] Corliss | A History of the Deep Space Network[END_REF]. The total number of spins in pink ruby crystal with slightly more than 0.05 percent Cr 3+ is about 2.5×10 25 per cubic meter. A magnetic field strength of 2500 gauss (G) is applied to the ruby in a direction that is perpendicular to the c-axis. The four ground-state microwave spin levels that occur are called paramagnetic levels or Zeeman levels. The population inversion (maser signal) between levels 2 and 1, resulting from pumping between levels 1 and 4.

By repeating the previous calculation, but respecting the new conditions, we find the maximum power of the signal maser :

P s max = lim P p →∞ P s = 4C 9τ 1 w S 0 N 24 -N 12 1 + x (22)
x = 0 f or P sc (Without Coherence) For different ions in crystals, we present in Table 1 the values of the spin-lattice and spin-spin relaxation times, the concentrations, the parameter x, the calculated and experimental powers. In that case, the coherences will have a large effect on the maser functioning as showed by the parameter x.

CONCLUSION

In this paper, we apply a full quantum model to the paramagnetic impurities that can used to design a maser source based on a high-quality sapphire monocrystal. We show that the maser power decreases as the coherence time τ 2 increases. The rate equation model predicts that the maser power linearly with paramagnetic dopant concentration N. Our model including coherence predicts a more realistic behavior for the maser power that converges to a limit independent of N.

FIGURE 2 .

 2 FIGURE 2. Schematic of stimulated emission at 12.04 GHz.

FIGURE 3 .

 3 FIGURE 3. Evolution of the maximum output power (nW) versus concentration for three values of τ 2 : 3 ns, 10 ns and 20 ns. τ 1 was assumed to be 10 ms.

FIGURE 4 .

 4 FIGURE 4. Evolution of the Threshold pump power versus concentration.

FIGURE 5 .

 5 FIGURE 5. Maser output power as a function of the output coupling β out for three resonators.

2Cτ 2 9τ 1 (N 14 +

 2114 N 24 ) f or P ac (With Coherence)In this case, the Bohr frequencies are ν 12 = 2.398GHz, ν 13 =12.88GHz and ν 14 = 24.44GHz. At thermal equilibrium (T= 4.2K) we have : N 12 = 0.0075, N 13 = 0.038 and N 14 = 0.068. The values of the transition parameters σ 2 i j for Cr 3+ paramagnetic ion in the Ruby are σ 2 12 = 1.59, σ 2 13 = 0.59 and σ 2 14 = 0.076.

  δ 12 e iω 21 t ; ρ 13 = δ 13 e iω 31 t ; ρ 23 = δ 23 e iω 32 t

TABLE 1 .

 1 Application of the model on other ions.

	Ion in crystal	τ 2 (ns)	τ 1 (ms)	N(ppb)	x	P sc (dBm) P ac (dBm) P exp (dBm)
	Fe 3+ in Pinocchio	2	18	16	0.0043	-52.03	-52.05	-53 [15]
	Fe 3+ in Geppetto	5.9	7	150	0.5	-35.36	-37.13	-37 [26]
	Cr 3+ in Ruby	60 [35]	157 [36] 500000	605	-10.43	-38.26	-40 [37]
	Mo 3+ in Sapphire >21 [38] <0.1 [39]	300	>500	-	-	-


Mohamad Mrad
email: mohamadmrad84@gmail.com

Ahmad Tarhini
email: tarhini.ahmad@gmail.com

Vincent Giordano
email: vincent.giordano@femto-st.fr

Absorbance and energy

Keywords: Energy-levels, Molecular Hamiltonian, Transition parameter, Absorbance, Shuffled complex evolution, Maser effect, State equations

published or not. The documents may come

I. INTRODUCTION

Masers based on paramagnetic ions hosted in a crystal have been widely studied and used as stable microwave sources or as low noise amplifiers in the 60's [START_REF] Siegman | Microwave Solid State Masers[END_REF]. The Maser action was obtained at low temperature in different doped crystals, the most popular of which was sapphire doped with Cr 3+ or Rubis. Many types of maser have been designed generally from a doped crystal inserted in a waveguide or from a metal cavity cooled to the temperature of liquid helium. The doping level was typically around 0.05% and for an X-band source, the cryogenic resonator exhibited a Q-factor of the order of 10000 or less. Most systems were based on the multiplicity of ground state levels to obtain the population inversion required by the action of microwave pumping. Some attempts to optically pump the crystal has been made [START_REF] Gayga | [END_REF]3]. But no real advantage over the classic Maser scheme has been demonstrated. The absence of a narrow band and an easy-to-handle laser source as well as the poor quality of the doped crystals available were certainly the main difficulties in obtaining a reliable optical pumped microwave maser. Finally, interest in this technology was almost stopped after the development of low noise cryogenic semiconductor amplifiers, stable quartz oscillators and low noise frequency synthesis.

More recently, we reported the possibility to get a 12 GHz maser signal presenting a short term relative frequency instability below 1 × 10 -14 at short term with a large marge of progress [4,5]. Such a frequency stability performance is required for a number of very demanding applications as metrology, space navigation, radio astronomy or fundamental physic experiments. The search for a reliable microwave source presenting a relative frequency stability of some 10 -15 is a challenge that motivates a lot of innovative works all around the world [START_REF] Giordano | Proc. of the 3rd International Colloquium -Scientific and Fundamental Aspects of the Galileo Programme[END_REF][START_REF] Kessler | Proc. of the 2011 Joint Conf. of the IEEE International Frequency Control Symposium and The European Frequency and Time Forum[END_REF][START_REF] Thorpe | [END_REF]. In our first prototype the sustaining amplification is achieved in a cryogenic sapphire whispering gallery mode resonator containing a small amount of paramagnetic F e 3+ ions. The F e 3+ ion exhibits three ground state energy levels. A population inversion is obtained between the two lower ground state levels by submitting the F e 3+ ions to a 31 GHz signal. The maser oscillation takes place at 12.04 GHz. As the sapphire resonator exhibits at low temperature extremely low losses, the F e 3+ ion concentration required to get enough gain is very low. Contrary to the maser of the 60's, our WhiGMO (Whispering Gallery Maser Oscillator) incorporates a microwave resonator presenting a Q-factor of typically one billion at 4.2K made from a sapphire crystal containing typically 0.1 ppm or less of active ions. The experimental validation of the 12 GHz WhiGMO and its potentiality to be competitive with the state-of-the-art microwave stable references opens the way to revisit the Maser concept and the associated material properties. Moreover the WhiGMO principle can be extended to other transitions of the F e 3+ ion in sapphire or to other doped crystals to design stable sources at higher frequencies.

With this in mind, in the current context, starting from the above introduction, this paper is organized as follows:

In section II we introduce the different parts of the molecular hamiltonian, the methods of calculation and the global optimization method based on Shuffled Complex Evolution Model [10]. Then in section III we explain the transition probability between two states using the expression of the dipolar transition moment operator. In section IV we shall explore and discuss the absorbance of a material with related calculations. Then in section V, we show the 31 GHz maser proposal by explaining the corresponding rate equations, signal power and the resonator design. Section VI is devoted to our conclusions.

II. THE MOLECULAR HAMILTONIAN

The F e 3+ electronic configuration is [Ar]3d 5 . The fundamental state spectroscopic term is 6 S (L= 0 and S= 5/2). The F e 3+ ion, in a doped sapphire crystal, replaces the Al 3+ cation and it is subjected to the action of the crystal field of closest ions O 2-and Al 3+ (figure 1). The cation is located in a distorted octahedral. If we assume that the cation F e 3+ is in symmetry with O h , then the fundamental term is 6

A 1g (O h ).
From the articles of Racah [12], Eliot and Stevens [13] and Judd [START_REF] Judd | Second Quantization and Atomic Spectroscopy[END_REF], the interpretation of the theory of an electronic spectra of transition metals became easy. The molecular Hamiltonian is given by:

H = H ee + H so + H c + H Zeeman (1) + T rees correction + Racah correction
We will discuss in this section the energy levels of a transition ion metal in a crystalline solid. The theory is already discussed and determined in detail in [START_REF] Judd | Second Quantization and Atomic Spectroscopy[END_REF][START_REF] Judd | Operator techniques in atomic spectroscopy[END_REF][START_REF] Wybourne | Spectroscopic properties of rare earths[END_REF][START_REF] Abragam | Electron paramagnetic resonance of transition ions[END_REF][START_REF] Liu | Spectroscopic properties of rare earths in optical materials[END_REF]. The different parts of the hamiltonian are: the electrostatic Coulomb interactions (H ee ), the spin-orbit coupling (H so ), the ion-ligand interactions described in the case of crystal field theory (H c ), the Zeeman interaction, and the Trees and Racah corrections.

A. Free ion

If we use the method of the "double-zeta exponential", and we assume that the radial function of d orbital can be written as:

R d (r) = a 1 ( (2ξ 1 ) 7 6! ) 1/2 r 2 exp(-ξ 1 r) + a 2 ( (2ξ 2 ) 7 6! ) 1/2 r 2 exp(-ξ 2 r) . (2)
The relation between the previous parameters is:

a 2 1 + a 2 2 + 2a 1 a 2 ( 2 √ ξ 1 ξ 2 ξ 1 + ξ 2 ) 7 = 1
We still have 3 independent parameters (a 1 , ξ 1 , ξ 2 ). From equation (2),the parameters of Racah B 0 and C 0 , the spin-orbit coupling constant ζ 0 , and the mean values r k 0 are calculated for the free ion (Zhao [19]). We don't forget to take the Trees [20] and Racah [21] corrections. The Trees correction constant α 0 = 40 cm -1 and the Racah correction constant β 0 = -131 cm -1 are fitted from the spectra of free ion [START_REF] Zhao | [END_REF].

B. Interaction of crystal field

The crystal field potential can be expressed as a linear combination of irreducible tensor operators C k q [START_REF] Wybourne | Spectroscopic properties of rare earths[END_REF]:

H c = kq B kq C k q (3)
where B kq are the parameters of crystal field. For d -electrons case, the k integer values can be 0, 2, 4 and -k ≤ q ≤ +k.

The equation (3) structure is given by the symmetry of the crystal lattice. For a (C 3v ) trigonal field, the summation has only four terms: B 20 , B 40 , B 43 and B 4-3 . For the case of a generalized crystal field model, the parameters B kq were given in [11]. The elements of the reduced matrix of tensor operators are given in the book of Nielson and Koster [START_REF] Nielson | Spectroscopic Coefficients for the p n , d n and f n Configurations Cambridge[END_REF] for 16 × 16 matrices. Using Wigner-Eckart theorem, we develop our generalized model based on 252 × 252 matrices.

C. Methods of calculation

Optimized parameters

The interaction term between ligand orbital and central metal ion is complex, so we use an approximation for electrostatic parameters B and C, the spin-orbit constant ζ, the corrections parameters α and β and the average values r 2 and r 4 :

B = N 4 B 0 , C = N 4 C 0 , ζ = N 2 ζ 0 r 2 = N 2 r 2 0 , r 4 = N 2 r 4 0 α = N 4 α 0 , β = N 4 β 0
where N represents the average reduction factor due to the covalency.

In the Al 2 O 3 crystal, the position of Al 3+ cation is defined respectively to the sites of the two nearest cations as shown in the figure 1. The inter-site distances and the corresponding angles with respect to the axis of symmetry C 3 are given by [START_REF] Macclure | [END_REF]24]:

R 0 1 =1.966 Å , θ 0 1 = 47.44 o R 0 2 =1.857 Å , θ 0 2 = 62.70 o
Due to the substitution of Al 3+ by F e 3+ , we represent the crystal distortion by assuming, as proposed by Zhao [25], that the two distances is affected by an identical relative variation. We therefore add an identical constraint to the angle relative variations. We then introduce two independent parameters f and g which can be defined as:

f = R 1 R 0 1 = R 2 R 0 2 ; g = θ 1 θ 0 1 = θ 2 θ 0 2
Finally the six independent parameters a 1 , ξ 1 , ξ 2 , N , f and g have been modified to approach the experimental observations.

Global Optimization Method (GOM)

At Arizona University (SCE-UA), they developed the shuffled complex evolution method [9,10] which has many advantages. It's a very effective and efficient global optimization method and it can be used to solve problems in non-linear physics with high-parameter dimensions. The aim of this algorithm, developed by Duan in 1992, is to reduce the problem into a best single parameter set in the feasible space. In the current work, this (SCE-UA) method is used to optimize the parameters of the molecular hamiltonian. In other words, it helps to obtain the energy levels (eigenvalues and eigenstates) of the hamiltonian. We want to apply this method in spectroscopy field. We have simplified the problem by using a correlation between different parameters in the previous section. Now, we use a general method: the system has six independent parameters to be optimized.

The optical spectrum of F e 3+ in α -Al 2 O 3 is conveniently represented by choosing:

a 1 = 0.677 , ξ 1 = 5.6 , ξ 2 = 1.723 N = 0.863 , g = 0.9958 , f = 1.003 (4)
In Table I, we show the comparison between experimental observations and our calculations. We also compared these results with Zhao's values [11]. We deduce that the introduction of the three free ion parameters and the sixth independent adjustable parameter (g) shows a better results in finding almost all the optical wavelengths in the F e 3+ ion spectrum.

To approve our physical description, we derived from our model the spin-Hamiltonian parameters that describe the fundamental state 6 A 1 (S) of the F e 3+ ion. Without applying a static magnetic field, there is three energy levels labelled in the following |i with i = 1, 2 or 3 represented in figure 2. Using the spin hamiltonian expression given in [START_REF] Bleaney | Reports on Progress in Physics[END_REF], the energy difference between the three hyperfine levels of the ground state are given by:

E 3 -E 1 = 1 3 [(18D + a -F ) 2 + 80a 2 ] 1/2 E 2 -E 1 = 3 2 (a -F ) -D + 1 6 [(18D + a -F ) 2 + 80a 2 ] 1/2 .
The values of the spin hamiltonian calculated parameters: D, a, a -F , and the gaps of energy are presented in Table II. They are compared to the Zhao's previous evaluation and the experimental values.

III. PROBABILITY OF TRANSITION

The transition between 2 states i and j, coming from the interaction with an electromagnetic dipole, is proportional to the square of the mean value of dipolar transition moment given as follow: The operator expression of the dipolar transition moment is given by:

M i→j = ψ j * μψ i dτ ≡< ψ j |μ|ψ j > . ( 5
)
μ = er. E + β( L + 2 Ŝ). B . (6)
The terms in equation 6 correspond to the electric dipole operator and the magnetic dipole, respectively. In general, the interaction between electric dipole with the light electric field dominates the transition by 4 orders of magnitude the interaction of magnetic dipole with magnetic field. The optical spectrum of the iron doped sapphire shows a quite intense absorption bands [START_REF] Lemann | [END_REF]30]. Thus it indicates the possibility of optical interaction with the F e 3+ ions. Our goal is to show the possibility to get a population inversion in the ground state by applying an optical radiation to the crystal. Thus we need to calculate the effect of an optical radiation on the population of each ground state level |i , with i = 1, 2 or 3. We have to take into consideration that the allowed transitions with different excited levels |j are listed in the table I. The transition rate W ij between one ground state level |i and one excited level |j is given by (Siegman [START_REF] Siegman | Microwave Solid State Masers[END_REF]):

W ij = W = 1 4 (γH) 2 g(ν)σ 2 (7)
where γ is the gyromagnetic factor, H is the module of the RF or the optical magnetic field and g(ν) is the line shape function describing the spectral broadening of the absorption line.

The dimensionless factor σ 2 represents the strength of the probability of transition. It depends on the relative orientation of the oscillating magnetic field with respect to the crystal axis z. σ 2 is computed from the matrix element of the interaction hamiltonian, which depends on the total magnetic momentum and the applied oscillating magnetic field. In our case, two different orientations have to be considered since we don't have a static magnetic field: H ⊥ z or H//z. We found that, for each transition, only one configuration gives noticeable value for σ 2 . In Table III, we give the relevant σ 2 ij parameters for each transition. The relative orientation of H with respect to z is indicated by the symbol (⊥) or (//).

IV. ABSORBANCE

In spectroscopy domain, the absorbance (also called optical density) of a material is a logarithmic ratio between the falling and transmitted radiation through this material. Thus, A = lg(P 0 /P trans ) [START_REF] Thorpe | [END_REF] where P 0 is the power of the incident radiation and P trans is the intensity of the transmitted radiation from the material. So we can decompose P 0 into two parts: P 0 = P abs + P trans where P abs is the total power absorbed by the ions. The power per unit volume can be written as: dP abs = hνW ∆ndV From equation 7, the transition rate is proportional to the square of the transverse magnetic field. According to the Pointing vector, this rate is proportional to the power of the applied wave (This is done in the Appendix A of Mrad's thesis [START_REF] Mrad | Effets Maser et CPT des ions paramagnétiques[END_REF]).

After an integral calculation, the relation between absorbed and transmitted powers in a waveguide of length L can be expressed as:

P abs (L) = P 0 (1 -e -αL ) = P 0 -P trans (9)
-n ). O h σ 2 1j σ 2 2j σ 2 3j 6 A1(S) 2.00 (0) [⊥] - - 3.27 (3) [⊥] 1.25 (0) [⊥] -
where α is the absorption coefficient, its expression is given as:

α = hγ 2 g(ν)∆n 2Z T νσ 2
where Z T represents the characteristic impedance of the guide. We put equation 9 in equation 8 in order to obtain the absorbance A:

A = αL
At the end, we conclude that A is proportional to the product νσ 2 , so to νσ 2 . In the figure 3, we plot the absorbance and the product νσ 2 versus wavenumber ν (unit cm -1 ). [START_REF] Lemann | [END_REF] and the product νσ 2 versus wavenumber ν

FIG. 3: Comparison between experimental absorbance

V. 31 GHZ MASER PROPOSAL

Laser sources emitting some 100 mW at 555 nm, with a linewidth less than 0.1 nm, are currently commercially available (see for example [31]). We select from the table III, the transition to the 4 E(G) excited level, at 18015cm -1 (≈ 555 nm). The transition probabilities for the 6 A 1 (S) -→ 4 E(G) transition with H ⊥ z are: σ 2 1 = 1.40×10 -3 , σ 2 2 = 1.80×10 -4 and σ 2 3 ≈ 0. So if we submit the crystal to a narrow band 555 nm laser source, the |1 level will be more effectively pumped. Therefore we can obtain large population differences between |1 and the two other ground state levels.

A. Rate equations

If we consider a F e 3+ doped Al 2 O 3 crystal submitted to a 555 nm narrow band source resonant with the transition 6 A 1 (S) -→ 4 E(G). The proposed system is schematized in figure 4. A cylindrical cryogenic sapphire resonator presenting a high-Q resonance mode at 31 GHz is submitted to an optical radiation at 555 nm. The crystal axis z is collinear to the cylinder axis. The 31 GHz resonant mode is assumed to be quasi-transverse magnetic, i.e. the magnetic field lines, in the equatorial plane of the cylinder, are perpendicular to the z axis. A small magnetic loop placed near the dielectric cylinder is used to derive the 31 GHz maser signal output.

The excited levels decay quickly to the ground state with a lifetime τ 0 1 ms. All relaxation in the ground state are governed by the spin-lattice effect with a characteristic time τ 1 ≈ 10 ms at 4 K. It is also determined that the spin-to-spin relaxation is very fast with a characteristic time τ 2 ≤ 20 ns. If we neglect coherence and we deal only with level populations, so a model with a simple rate equations is sufficient to describe the interaction [32]. There is a possibility that the excited ions may fall on intermediate levels, but since τ 0 τ 1 they relax rapidly to ground state. This fact leads to consider only a four levels system: the three ground state levels and the 4 E(G) excited one, which is labelled |4 . The absorption rate W P due to excitation of the ground state ions from the optical radiation is evaluated using equation 7. The optical pumping gives the possibility to get a population inversion between |3 and |1 . Therefore one can obtain a maser effect at 31 GHz provided that the resonator has been designed to exhibit a high Q resonance at this frequency. The first 31 GHz emitted photons will stimulate others emissions with a rate W S . If the optical pumping is efficient enough to compensate for the loss of the resonator,a self-sustaining 31 GHz oscillation will occur. We have followed the method described in [33] (validated in case of 12 GHz WhiGMO), in order to determine theoretically the conditions to get a 31 GHz maser signal. The stationary solutions of the rate equations are:

∆n 13 = 27∆N 13 + 6(2∆N 13 -3∆N 14 )τ 1 W P 9(3 + 4τ 1 W S ) + 4(3 + τ 1 W S )τ 1 W P (10a) ∆n 14 = 9(3∆N 14 + 2(2∆N 14 -∆N 13 )τ 1 W S ) 9(3 + 4τ 1 W S ) + 4(3 + τ 1 W S )τ 1 W P , (10b)
where ∆N ij and ∆n ij are the difference in population between two levels |i and |j at the thermal equilibrium and in the presence of the pump and maser signal, respectively. ν 14 and ν 13 represent the frequencies of the pump and the maser signal respectively.

B. Threshold pump power

When the population inversion takes place, the threshold pump power is achieved in the absence of the maser signal. By taking ∆n 13 = 0 in equation (10a), we can find the absorption rate W P0 corresponding to this situation:

W P0 = 9 2τ 1 ∆N 13 3∆N 14 -2∆N 13 (11)
The total absorbed power can be written as:

P 0 = hν 14 W P0 ∆n 140 N V ef f (12) = 3 2τ 1 hν 14 ∆N 13 N V ef f
where N is the concentration of F e 3+ ions, and V ef f is the volume occupied by the active ions which participate to the maser signal. V ef f is the volume of the 31 GHz mode in the sapphire resonator (see section V D). Here, we assume that the optical radiation only illuminates the resonator part where the 31 GHz mode is concentrated. P 0 is thus the minimal optical power required to get maser action in optimized conditions.

C. Maximum maser signal power ∆n 13 , and thus the 31 GHz maser signal power, increase with the pump signal power system until the saturation of the optical transition occurs, i.e n 4 n 1 . Assuming we already overcome this situation, i.e. W P → ∞, equations (10a-10b) will simplify and the maximum maser power is calculated to obtain:

P 31max = hν 13 M ax[W S ∆n 13 ]N V ef f (13)
= 3 2τ 1 hν 13 (3∆N 14 -2∆N 13 )N V ef f P 31max is the total power generated by the active ions inside the resonator. The coupling loop derive only a fraction of this power depending on the coupling coefficient β, which is related to the loop area and to its position with respect to the sapphire resonator. For a critical coupling , i.e. β = 1, the maser output power is one half of P 31max . The resonator Q-factor is also impacted by the loading to the external circuit. For β = 1, the loaded Q-factor is one half of the unloaded one.

D. Resonator design

Our objective is to use the F e 3+ ions present as contaminant in a high purity sapphire monocrystal. Typical effective ions concentration in such a type of crystal is well below 1 ppm. Thus the 31 GHz resonator has to present a very high quality factor, otherwise the 31 GHz stimulated emissions will not be sufficient to compensate for the resonator loss. This can be accomplished by using a whispering gallery mode, which Q-factor is only limited by the sapphire losses. Quasi-TM Whispering gallery modes are characterized by three integers m, n, l which represent the electromagnetic field component variations along the azimuthal, radial and axial directions respectively [34]. In the following we consider only the resonant modes with low radial and axial variations, i.e. those corresponding to n = l = 0. At low temperature, i.e. near the liquid helium temperature, the whispering gallery mode Q-factor can be as high as 1 billion providing the mode order is sufficiently high, i.e. m > 15 [35]. The WhiGMO principle has been demonstrated at 12 GHz, i.e. the frequency of the |1 → |2 transition. In this case, the population inversion results from a pumping at 31 GHz. The resonator has a diameter 2R = 50 mm and a thickness h = 30 mm. The quasi-TM Whispering Gallery mode W GH 17,0,0 is used to support the maser signal [36].

For the whispering gallery mode the electromagnetic fields are concentrated near curved air-dielectric interface, between the resonator radius R and a caustic of radius R C . The effective volume

V ef f = πh R 2 -R 2
C occupied by the mode is a little mode dependant as R C is given by:

R C = m r ν 13 c 2 - π 2h 2 (14)
where r ≈ 9.4 is the sapphire permittivity is the transverse direction with respect z. c the velocity of light in vacuum.

To design a 31 GHz resonator we keep the same ratio diameter/thickness, which is near the optimal value [37]. Thus to design the cryogenic resonator only means choosing the azimuthal number for the mode supporting the 31 GHz signal. m has to be higher than 15 to prevent radiation losses which limit the Q-factor and should not be to high otherwise the coupling with the output probe will be difficult to adjust. The table IV gives for different values of m, the resonator diameter 2R, the threshold optical power P 0 and the maximum maser power emitted by the ions. Here, the F e 3+ concentration is assumed to be 0.02 ppm, i.e. the same value than the 12 GHz WhiGMO. In this work we refined the description of the F e 3+ ion embedded into a (α-Al 2 O 3 ) matrix to derive its 252 energy levels with two optimization methods "local and global", whose the global method is based on Shuffled Complex Evolution Model. Moreover we calculated the F e 3+ ion strength transition probability including those for optical transitions. It appears than a few allowed optical transitions are favorable to obtain a maser operation at 31 GHz applying optical pumping at wavelength achievable with available low cost laser sources. Eventually, a first preliminary evaluation of the 31 GHz maser concept is presented. We have also shown that the 31 GHz maser signal power achievable is larger than the classical 12 GHz version, i.e. a power gain of 100 is expected in the optimized conditions. As fundamental limit of frequency stability is fixed by the thermal noise, any increase in the maser signal output will benefit to the performance.

FIG. 1 :

 1 FIG. 1: Local structure distortion of octahedral F e 3+ center in α-Al2O3 system.

FIG. 2 :

 2 FIG.2: Structure of F e 3+ ground state in the α-Al2O3 crystal without applied static magnetic field.

FIG. 4 :

 4 FIG.4: Energy levels and schematic of the cryogenic optically pumped 31 GHz WhiGMO.

TABLE I :

 I Calculation and observation of the d-d transition for the crystal α-Al2O3 : F e 3+ (Values are in cm -1 ).

	O h Zhao [11] Mrad Observed [27]
	4 T1(G)	9790	9700 9450 -9700
		10048	9990
		10229 10128
		10392 10294
		10473 10381
		10493 10396
	4 T2(G) 14229 14183	14350
		14262 14208
		14315 14219
		14373 14256
		14399 14281
		14401 14282
	4 A1(G) 17977 17875 17600 -17800
		17998 17897
	4 E(G) 18101 18015
		18143 18056
		18188 18105
		18199 18113
	4 T2(D) 20181 20075
		20191 20078
		20341 20233
		20533 20414
		20562 20432
		20823 20470
	4 E(D) 22299 22195 22120 -22200
		22313 22212
		22319 22219
		22328 22229
	4 T1(P ) 25652 25663 25680 -26700
		25805 25755
		25945 25875
		26141 26090
		26330 26225
		26457 26526
	4 A2(F ) 30167 29884	29000
		30170 29887

TABLE II :

 II Calculated spin-Hamiltonian parameters for α-Al2O3 : F e 3+ at 300 K compared to experimental values (the unit is 10 -4 cm -1 ).

	ZFS	Zhao[11] Mrad Observed[28, 29]
	parameters
	D	1784	1723	1718 ± 2
	a-F	322	330	329 ± 2
	a	254	268	241 ± 4
	E3 -E1	10838 10478	10451
	E2 -E1	4118	4011	4015

TABLE III :

 III Strength of transition probability σ 2 for the F e 3+ d-d transitions in α -Al2O3. The integer (n) in brackets is for ×10

TABLE IV :

 IV Different resonator designs to support the 31 GHz signal. The pump threshold power P0 and the maximum maser power P31 max are given for few azimuthal numbers as well as the resonator diameter assuming a constant ratio 2R

	VI. CONCLUSION
			h = 5 3 .
	m 2R	P0 P31 max
	(mm) (mW) (µW)
	17 19.22 1.17	1.02
	19 21.13 1.42	1.24
	21 23.05 1.71	1.49
	25 26.88 2.35	2.05
	35 36.46 4.40	3.85
	49 50.00 8.41	7.36

T1(G) 1.84[START_REF] Giordano | Proc. of the 3rd International Colloquium -Scientific and Fundamental Aspects of the Galileo Programme[END_REF] [//] 7.43(5) [⊥] 1.70[START_REF] Giordano | Proc. of the 3rd International Colloquium -Scientific and Fundamental Aspects of the Galileo Programme[END_REF] [⊥]


Valérie Declerck

Ariel Pérez-Mellor

Régis Guillot

David J Aitken

Michel Mons

Anne Zehnacker

Vibrational circular dichroism as a

Keywords: VCD, β peptide, crystal structure, IR spectroscopy

Introduction

Synthetic oligomers built by assembling particular molecular building blocks can adopt regular folding patterns, which are maintained by networks of non-covalent interactions such as hydrogen bonds, locking them into well-defined conformations. The environment and the nature of the building blocks dictate the shape of these oligomers, leading to diverse secondary structures similar to those found in Nature (such as helices turns and sheets) as well as abiotic molecular architectures. In the emerging area of foldamer science, bioinspired β-amino acids (β-AA) are of special interest due to their pronounced ability to induce regular folding patterns in oligomer sequences, including helical structures in particular. Moreover, oligomers of β-AAs, also known as β-peptides, are resistant to proteolysis which is especially interesting for potential biomedical applications. 1 In βpeptides, local hydrogen bonding interactions compete with long-distance interactions. In this context, employment of cyclic β-AA monomers which incorporate a 4-to 6-membered aliphatic ring is of special interest because the monomer can possess either a cis or trans geometry and the geometric constraints imposed by the ring induce a pronounced conformational bias. 2 3 The 4-membered ring monomer, 2-aminocyclobutane carboxylic acid (ACBC), displays a preference for local interactions implicating a six-membered (C6) or an eightmembered (C8) cyclic NH•••O hydrogen bond, depending on the cis or trans ring geometry, the physical or chemical environment, and the length of the oligomer. 4 Several studies have been conducted in solution to assess the conformational preferences of these systems. In derivatives of the cis stereoisomer (cis-ACBC), the intra-residue sixmembered ring (C6) hydrogen bond suggested for the monomer in solution seems to persist in oligomers up to the octamer. 5 6 However, the presence of a C8 monomeric structure cannot be ruled out in the condensed phase. Derivatives of the trans diastereoisomer (trans-ACBC) display a C8 hydrogen bond in the monomer and in short oligomers (up to four residues), which is superseded by 12-helix structures, implicating a series of C12 hydrogen bonds, in longer oligomers. Fewer studies have been reported under isolated conditions. Pohl et al. reported an Abstract: Peptide models built from cis and trans 2aminocyclobutane-1-carboxylic acids (ACBCs) are studied in the solid phase by combining FTIR absorption spectroscopy, vibrational circular dichroism (VCD), and quantum chemical calculations using density functional theory (DFT). The studied systems are N-tert-butyloxycarbonyl (Boc) derivatives of 2-aminocyclobutanecarboxylic acid (ACBC) benzylamides, namely:

Boc-(cis-ACBC)-NH-Bn and Boc-(trans-ACBC)-NH-Bn. These two diastereomers show very different VCD signatures and intensities, that of the trans-ACBC derivative being one order of magnitude larger in the region of the (CO) stretch. The spectral signature of the cis-ACBC derivative is satisfactorily reproduced by that of the monomer extracted from the solid-state geometry of related ACBC derivatives, which shows that no long-range effects are implicated for this system. In terms of hydrogen bonds, the geometry of this monomer is intermediate between the C6 and C8 structures (exhibiting a 6-or 8membered cyclic NH•••O hydrogen bond) previously evidenced in the gas phase. The benzyl group must be in an extended geometry to reproduce satisfactorily the shape of the VCD spectrum in the (CO) range, which qualifies VCD as a potential probe of dispersion interaction. In contrast, reproducing the IR and VCD spectrum of the trans-ACBC derivative requires clusters larger than four units, exhibiting strong intermolecular Hbonding patterns. A qualitative agreement is obtained for a tetramer, although the intensity enhancement is not reproduced. These results underline the sensitivity of VCD to the long-range organisation in the crystal.

elegant study of simple ACBC amide derivatives under matrix isolation conditions. 7 They provided evidence for both C6 and C8 geometries in the cis-ACBC monomer. Surprisingly, a fully extended structure with no hydrogen bond coexisted with the expected C8 structure for the trans-ACBC stereoisomer. We recently applied conformer-specific laser spectroscopy to jetcooled ACBC benzylamides, protected by a tertbutyloxycarbonyl (Boc) group on the N-terminus; the benzyl group was used as the requisite UV chromophore for conformerspecific spectroscopic analysis. 8 The gas-phase results confirmed the condensed-phase findings that C6 and C8 conformers both exist for monomeric cis-ACBC derivatives. 8 A structure involving an outstandingly strong C8 hydrogen bond was observed in the jet-cooled monomeric trans-ACBC derivative. 8 The conformational preferences observed in isolated species, whether in gas phase or in solution, may be modified in the solid state due to intermolecular hydrogen bond formation. In hexamers or octamers of trans-ACBC, the intramolecular C12 interaction is not perturbed in the crystal lattice and the 12-helical secondary structure is similar in solution and in the solid state. 9 For shorter oligomers, however, it is expected that the intramolecular hydrogen bonds are disrupted by the formation of intermolecular hydrogen bond networks, as observed in cis-ACBC. 10 Vibrational circular dichroism (VCD) has proved to be a powerful tool for the study of secondary structures in peptides and proteins. 11 12 12d, 13 14 The Amide I and II regions are especially informative as the coupling between modes localised on CO or NH of different residues has a strong signature in VCD. 15 In contrast with electronic circular dichroism (ECD) signatures, which has been used for structural characterization of helices or strands, 2,16 VCD has not been applied yet to ACBC-based peptides, in particular to their solid state. VCD spectra of solidstate or supramolecular systems, 17 for example nanofibrils, 18 polymorphic crystals, 19 or foldamers 20 are scarce. One of the questions raised by the interpretation of VCD spectra in the solid state is the nature of the minimum size cell that is necessary for describing the system. The unit can be thought of as a monomer, like in solution. [START_REF] Barbu-Debus | Physical chemistry chemical physics[END_REF] In other cases, one has to take into account the crystal unit cell itself or a cluster of definite size like a dimer or a tetramer. 15c 22 Still, the intensity enhancement observed in the solid state has to be fully understood. 18, 23 24 We present here a comparative FTIR and VCD study of the solidstate cis-and trans-ACBC derivatives, Boc-(cis-ACBC)-NH-Bn and Boc-(trans-ACBC)-NH-Bn, shown in Figure 1. These ACBC derivatives are protected by a tert-butyloxycarbonyl (Boc) group on the N-terminus and bear a benzyl amide (Bn) group on the C terminus. In what follows, Boc-(cis-ACBC)-NH-Bn and Boc-(trans-ACBC)-NH-Bn will be referred to as cis-ACBC and trans-ACBC, respectively, like in our previous publication. 8 We will focus on the vibrational spectroscopy in the amide I and II region (ν(CO) stretch and β(NH) bend region), which is very sensitive to hydrogen bond formation. The calculation strategy rests on the definition of clusters of finite size built from known X-ray structures. The IR absorption and VCD spectra are calculated for increasing sizes of the clusters until the calculations satisfactorily reproduce the experimental results, which allows determining the smallest cluster size that is necessary for reproducing the VCD spectrum of the crystal.

Materials and Methods

SYNTHESIS AND CHARACTERIZATION

Compounds (-)-Boc-(t-ACBC)-OH, and (-)-Boc-(c-ACBC)-OH were prepared according to literature procedures. 25 All reagents and solvents were of commercial grade and were used without further purification. Dichloromethane was dried over activated alumina. Flash chromatography was performed with SDS silica gel (35-70 μm). Analytical thin-layer chromatography (TLC) was performed with 0.25 mm commercial silica gel plates (EMD, Silica Gel 60F254). TLC plates were visualised by UV fluorescence at 254 nm then revealed using a phosphomolybdic acid solution (10% in EtOH). Retention factors (Rf) are given for such analyses. Fourier-transform Infrared (IR) spectra were recorded in the ATR mode by deposition of the neat compound on the ATR diamond accessory. Maximum absorbance (ν) is given for significant bands in cm -1 . Melting points were obtained in open capillary tubes and are uncorrected. Optical rotations were measured using a 10 cm quartz cell. Values for [α] D T were obtained with the D-line of sodium at the indicated temperature T, using solutions of concentration (c) in units of g

General Procedure for the Preparation of Benzylamides

To an ice-cold solution of Boc-amino acid (43.1 mg, 0.20 mmol, 1 eq.) in dry CH2Cl2 (10 mL) were added 1-hydroxybenzotriazole monohydrate (37.8 mg, 0.28 mmol, 1.4 eq.) and benzylamine (24 μL, 0.22 mmol, 1.1 eq.). The mixture was stirred for 10 min at 0 °C then 1-(3-dimethylaminopropyl)-3-ethyl carbodiimide hydrochloride (57.5 mg, 0.30 mmol, 1.5 eq.) was added. The resulting mixture was then stirred for 48 h at room temperature, then washed successively with 1 M aqueous KHSO4 (5 mL) and saturated NaHCO3 (5 mL). The organic layer was dried over MgSO4, filtered and evaporated under reduced pressure. Flash chromatography (CH2Cl2/EtOAc) gave the benzyl amide.

N-Benzyl-(1R,2R)-2-(t-butyloxycarbonylamino)cyclobutane carboxamide (Boc-(t-ACBC)-NH-Bn):

Coupling of (-)-Boc-(t-ACBC)-OH according to the general procedure, followed by flash chromatography (CH2Cl2/EtOAc = 80/20) gave the benzyl amide (-)-Boc-(t-ACBC)-NH-Bn (47.

N-Benzyl-(1R,2S)-2-(t-butyloxycarbonylamino)cyclobutanec arboxamide (Boc-(c-ACBC)-NH-Bn):

Coupling of (-)-Boc-(c-ACBC)-OH according to the general procedure, followed by flash chromatography (CH2Cl2/EtOAc = 90/10) gave the benzylamide (-)-Boc-(c-ACBC)-NH-Bn (59.

X-ray structures

Crystal data has been deposited to the Cambridge Crystallographic Database with CCDC number 1908274 (Boc-(c-ACBC)-NH-CB) and contain the supplementary crystallographic data for this paper. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif. 26

EXPERIMENTAL SPECTROSCOPIC METHODS

The vibrational IR absorption and VCD spectra were measured using an FTIR spectrometer (Vertex 70, Bruker) equipped with a VCD module (PMA 50, Bruker). The signal was measured by an MCT IR detector equipped with a BaF2 window and cooled with liquid nitrogen. A spectral resolution of 4 cm -1 was used for both absorption and VCD spectra. The IR radiation was polarized with a linear polarizer then modulated by a 50 kHz ZnSe photo-elastic modulator (Hinds). A low-pass filter cutting at 2000 cm -1 was added before the linear polarizer to increase the dynamical response of the detector. The signal of the MCT detector was demodulated using a lock-in amplifier (Stanford Research Systems SR 830). The alignment of the spectrometer was carefully verified by checking the mirror-image relation between the VCD spectra of the two enantiomers of camphor (0.3 M in CCl4). The samples were prepared by grinding 6 mg of the studied molecule with 2 g of KBr in a mixer mill (MM 400 Retsch) at 20 Hz during 1 h. The artefacts due to the birefringence of the KBr pellets were eliminated by following the procedure proposed by Merten,27 derived from that introduced by Buffeteau. 28 It consists in rotating the sample in the plane perpendicular to the light propagation axis for each side (front F or back B) of the pellet and averaging the spectra obtained for each position of the pellet at 0° and 90° (Equation 1).

I VCD (ν ̃) = 1 8 ∑ I VCD (ν ̃; n⋅90°; F) + I VCD (ν ̃; n⋅90°; B) 3 n=0 (1)
For the sake of better averaging and artefact control, the procedure was extended to 180° and 270°. The measurements were repeated on a second sample. The final spectrum was obtained by adding 1400 scans for each sample position and averaging all the positions so that the total acquisition time is ~5 h 30 for all the spectra shown below.

THEORETICAL METHODS

The calculation strategy encompassed the following steps, which rest on the X-ray structures. Single crystal X-ray diffraction data for cis-ACBC and trans-ACBC were not available for this study; instead, we used the X-ray data of the related compounds cis-ACBC* and trans-ACBC*, in which the benzyl amide (Bn) is replaced by a cyclobutyl amide (c-Bu), shown in Figure 1. Firstly, a supramolecular unit of the molecule under study was constructed from the crystal of the model structure. The hydrogen bond pattern was kept identical to that of the model system and the cyclobutyl is substituted by a benzyl. The preliminary structure obtained thereby was optimized using periodic boundary conditions (PBC). The PBC calculations were limited to onedimension along the b axis, i.e. the hydrogen bonded strands. Then a core of finite size was extracted from the resulting structure. The IR absorption and VCD spectra were calculated for increasing sizes of the core until the calculations satisfactorily reproduce the experimental results, which allows determining the smallest cluster size that is necessary for reproducing the VCD spectrum of the crystal. The RI B97-D/TZVPP -D3 calculations were performed using the Turbomole package. 29 All the other calculations were performed using the Gaussian 09 program. 30 Optimization of the preliminary structure using PBC was performed at the B3LYP/6-31G level of theory, with sixteen units per cell. A core of four molecules was then extracted from the PBC results and calculated at the RI B97-D/TZVPP -D3 level which is known to yield satisfactory frequencies for peptides at an acceptable computational cost. For the sake of testing the ability of simpler assembly to account for the experimental data, dimers and monomers were also extracted from this tetramer. The resolution of the identity was used for reducing the calculation time. 31 The harmonic frequencies were calculated at the same level of theory. Dispersion-corrected calculations, with the same basis set and functional, were used previously for the same molecules isolated in the gas phase and yielded excellent agreement with the experimental values. 8 The calculated harmonic frequencies were scaled in the 3 m region to correct them for anharmonicity and basis set incompleteness. The scaling factor used (0.96) was defined as the ratio between the experimental frequencies of the conformers observed in the gas phase and the harmonic frequencies calculated at the RI B97-D/TZVPP level. This value is found to be insensitive to the fact that dispersion corrections are used or not. No scaling factor was needed in the fingerprint region. Then, the VCD spectra were calculated for increasing sizes of clusters extracted from the tetramer, stopping at the smallest size that gives satisfactory agreement with the experiment. VCD calculations were carried out, with the previously obtained geometries, at the B3LYP/6-31G++(d,p) level for the monomers and 6-31G(d,p) level for the dimers or the tetramers. 32 The 6-31G++(d,p) level was chosen because it has been shown to give satisfactory results in calculating VCD spectra of flexible molecules. 33 Moreover, using the smaller basis set 6-31G(d,p) gives results similar to those at the B3LYP/6-31G++(d,p) level for cyclic dipeptides and allows calculating large systems at a reasonable computational cost. 15c Here also, similar results are obtained with the two levels of theory, as shown in Figure S1 on the example of the Mcis cis-ACBC monomer, extracted from the tetramer, and eventually used for the assignment (vide infra). In order to test the sensitivity of the VCD signals to dispersion interactions as will be discussed below, calculations have been carried out in the presence or absence of the D3 set of dispersion parameters. 34 When not specified otherwise, the reported calculations do not include dispersion. All the monomers observed in the gas phase and reported previously were recalculated here, including dispersion or not, for the sake of comparison. The harmonic frequencies were scaled by 0.96 in the 3 m region, i.e. the ratio between the experimental frequencies in the gas phase and the harmonic calculated frequencies, and 0.97 in the fingerprint region.

Results and Discussion

CRYSTALLOGRAPHIC DATA

The 1 and2.

GAS-PHASE SPECTROSCOPIC DATA$

cis-ACBC and trans-ACBC have been fully characterised in the gas phase by conformer-selective laser spectroscopy under supersonic expansion conditions. 8 In brief, the experimentally observed conformation of trans-ACBC has been assigned to a structure involving an 8-member ring (C8) bridging the N(10)H and C(3)O groups, named t-1. The C8 interaction is outstandingly strong and manifests itself by a broad red shifted feature at 3021 cm -1 assigned to the (NH) stretch. It should be noted that the most stable calculated structure t-1 involves interaction between the t-butyl group and the phenyl ring, a point to which we shall return in the discussion. 8 Two conformers have been experimentally observed for cis-ACBC. The most intense bands of the electronic spectrum have been assigned to a C8 structure, named c-1, while weaker electronic transitions have been attributed to a C6 structure, named c-3. Due to the close position of the two amide functions, only frustrated hydrogen bonds can be formed. As a result, the C8 interaction is much weaker in cis-ACBC than in trans-ACBC, with a (NH) stretch observed at 3414 cm -1 , compared to the free (NH) stretch observed at 3485 cm -1 . The C6 structure displays a weak C6 hydrogen bond, with bounded and free (NH) stretches observed at 3465 and 3480 cm -1 , respectively. Interestingly, a second conformer of the C8 form of both cis and trans-ACBC, namely t-2 or c-2, was calculated, with similar G at room temperature as t-1 or c-1, with similar H-bonding pattern but no interaction between the phenyl and t-butyl groups. It was supposed that this second conformer relaxes to the most stable one in the late expansion. It should be noted however that due to the similar hydrogen bond patterns of t-2 and t-1, and of c-1 and c-2, respectively, these forms could not be distinguished based on their IR spectra only.

SOLID-PHASE SPECTROSCOPIC RESULTS

cis-ACBC

The experimental IR absorption and VCD spectra of cis-ACBC in the fingerprint region are shown in Figures 3a and3c, respectively. The comparison between the VCD spectra of the two enantiomers is given in Figure S2. The IR absorption shows two bands in the Amide I region ((CO) stretches), located at 1681 and 1650 cm -1 . These bands appear as singlets of identical sign in the VCD spectra. The shape of the spectrum, in particular, an identical sign for the two (CO) stretches, suggests a structure in which the (CO) stretches are not coupled. The Amide II region ((NH) bends) displays two broader bands, at 1537 cm -1 and 1520 cm -1 , which appear as a bisignate doublet in the VCD spectrum. The presence of a doublet may indicate either coupled modes or two non-coupled vibrations with opposite rotatory strengths. 15c The intensity (difference in absorbance between left and right circular polarised light) of the bands is of the order of 10 -4 for the strongest features. The spectrum of a tetramer core calculated at the RI B97-D/TZVPP level including or not D3 corrections is compared to the experiment in Figures S3.

In both cases, the hydrogen-bonded strand tends to bend when the chain length increases, which results in non-equivalent hydrogen bonds. Calculations including dispersion tend to deform the structure more than those that do not include dispersion, which results in a less satisfactory agreement between experimental and simulated spectra, probably because the interaction with the adjacent strands is not taken into account. However, the hydrogen bond pattern in the core of the oligomer is similar whatever the method used. S4 of the supplementary information. The spectrum of the monomers present in the gas phase, namely c-1 and c-3, as well as that of c-2, are shown in Figure 3f, 3g, 3h, respectively, for the sake of comparison. These three structures are shown in Figure 4. The IR absorption spectrum of the most stable gas-phase structures qualitatively agrees with the experimental IR absorption spectrum but better agreement is obtained between the experiment and the spectra of Mcis. In particular, the spacing between the CO stretch frequencies is well reproduced in the simulated IR absorption spectrum of Mcis. The CO•••HN distances of this monomeric subunit are intermediate between that of the so-called c-1 and c-3 structures evidenced in the gas phase (see Table 1). Indeed, the N4H c1 structure evidenced in the gas phase, with a C8 intramolecular hydrogen bond, optimised without dispersion b2) same with dispersion c) c3 structure evidenced in the gas phase, with a C6 intramolecular hydrogen bond optimised without dispersion, c2) same with dispersion.

The experimental VCD spectrum, shown in Figure 3c, can be divided into two regions. The amide II region where the observed bisignate signature is satisfactorily reproduced by any of the structures mentioned above. It is neither sensitive to the nature of the intramolecular H bond nor to the position of the substituents. The bisignate doublet corresponds to the (N4H) and (N10H) bends, which display opposite VCD signs. The lack of sensitivity of the VCD signal of these modes to the environment and structural parameters is characteristic of what are called robust modes. 14, 35 36 Indeed, c-1, c-3, c-2, and Mcis show a similar bisignate pattern in this range, except for slight differences in intensities. Depending on the structure considered, the two (NH) bends are coupled (in c-3) or not (in Mcis) but the shape of the VCD spectrum in this range remains similar, with two bands of opposite signs. Also, examination of the structures shown in Figure 4 indicates that the relative position of the NH is not the same in the three systems. In c-1, the two NH make an angle of ~30°, while this angle is only of the order of ~5° in Mcis and close to zero in c-3. Including dispersion in the calculation or not does not alter the shape of the spectrum either. The (CO) stretch region displays much more variability. In c-1, the (CO) stretches are slightly coupled and a band of the wrong sign is calculated. In c-3 or Mcis, the (CO) stretches are completely localized and appear as signals of identical sign. In order to check whether the localized or delocalized character of the (CO) stretches really is what impinges the shape of the VCD spectrum, we have compared the calculated spectrum of c-1, which is the most stable conformer in the gas phase, to that of c-2. c-2 displays exactly the same hydrogen bond pattern as c-1 but differs in the position of the phenyl, which is interacting with the BOC substituent in c-1 but not in c-2. The IR absorption spectra of c-1 and c-2 are identical, as shown in Figure 3. The bisignate signature observed in the region of the (NH) stretches is similar in c-2 and c-1, although slightly more intense in c-1. The two conformers completely differ in the shape of their VCD spectrum in the region of the (CO) stretches. While c-1 displays a single band of the wrong sign, c-2 shows two bands of identical sign, as experimentally observed. The (CO) stretching modes are slightly coupled in c-2, like in c-1, but their VCD signatures in the (CO) stretch region are different. We can also examine the effect of including dispersion. For a given geometry, including dispersion in the frequency calculations does not modify the IR and VCD spectra. However, at the geometry optimization stage, dispersion tends to make the t-butyl and aromatic ring closer to each other, which in turn results in shorter NH…OC distances. As a result, the (CO) stretching modes, which are only slightly coupled in c-1 optimised without dispersion corrections, are fully delocalised when c-1 is optimised including D3 corrections. This change is also reflected in the shape of the VCD spectrum in the (CO) stretch region (see Figure S4). Dispersion also modifies the VCD spectrum of c-3.

When optimization is performed without dispersion, c-3 shows two bands of identical sign, but the low-energy one is so weak that it cannot be distinguished in the spectrum. When including dispersion in the calculations, two bands of comparable intensity are observed. This example illustrates the extreme sensitivity of VCD to subtle structural parameters, in this case, dispersion driven structural modifications. When including dispersion, the optimisation of Mcis results in a structure similar to c-1, hence an unsatisfactory VCD spectrum. We can conclude from these results that the presence of two bands of identical sign in the (CO) stretch region is the signature of an open structure, and can be taken as an indication of non-interacting t-butyl and phenyl ring. We can conclude that description of dispersion is of extreme importance here as overestimating dispersion results in a folded geometry and hence in wrong VCD spectrum in the (CO) stretch region.

trans-ACBC

The IR absorption and the VCD spectra of trans-ACBC in the fingerprint region are shown in Figures 5a and5e. The comparison between the VCD spectra of the two enantiomers is given in Figure S5. Only one absorption band appears in the β(NH) region, at 1535 cm -1 , a value characteristic of a bound NH group. This band is broader (18 cm -1 FWMH) than the other features observed in the IR absorption spectrum and probably corresponds to the superposition of β(NH) bending modes of different nature but similar frequencies. In contrast, two bands are observed in the ν(CO) stretch region, in ranges characteristic of the ν(C3O) (ester) (1684 cm -1 ) and ν(C9O) (amide) (1645 cm -1 ) stretches. Each (CO) and β(NH) band observed in absorption appears as a bisignate couplet in the VCD spectrum. The VCD spectrum is much more intense than that of cis-ACBC, with the strongest intensity, of the order of 10 -3 , being one order of magnitude larger than in cis-ACBC. However, the intensity in the β(NH) range is of the same order as in cis-ACBC, the major difference is observed in the (CO) range. We have followed the same procedure as for cis-ACBC and optimised the tetramer extracted from the PBC calculations at the B97-D3 level. As observed for cis-ACBC, including dispersion results to a folding of the hydrogen-bonded strand, hence poorer agreement between experimental and simulated IR spectra. Thus, we have favoured calculations without dispersion corrections. The simulated IR absorption deduced from calculations without dispersion corrections is compared to the experiment in Figure S6. Notwithstanding the boundary effects, the IR absorption spectra show satisfactory agreement with the experiment, in particular, the gap between the β(NH) and the (CO) is correctly reproduced.

The monomer extracted from the PBC calculations, called Mtrans in what follows, qualitatively reproduces the observed IR absorption, in contrast with the gas-phase most stable monomer t-1 for which the bound β(NH) is calculated much too high in energy relative to the experiment (see Figure 5). This suggests that the intramolecular C8 hydrogen bond observed in the gas phase is disrupted in the solid to the benefit of intermolecular hydrogen bonds. These intramolecular hydrogen bonds are not as strong as the intramolecular C8 interaction evidenced in the gas phase monomer. The spacing between the β(NH) bend and the (CO) stretch frequencies calculated in Mtrans is overestimated relative to the experiment. Moreover, there is no mean of describing the VCD spectrum in terms of a monomer only. Indeed, the fact that each absorption band appears as a bisignate doublet indicates that a cluster must be considered. However, the dimer does not reproduce the IR spectrum, in particular, the gap between the β(NH) bend and the (CO) stretch bands (see Figure S5). Moreover, the two (CO) stretch transitions appear in the region of bound (CO) and show a bisignate signature. This suggests that a structure with at least four bonded COs, i.e. a tetramer, should be taken into account. We therefore calculated the IR absorption and the VCD spectra of the tetramer, at the B3LYP/6-31G(d,p) level. To avoid consequences of boundary effects in the vibrational spectroscopy, the termini were isotopically modified. The hydrogen atoms of the two free NH were replaced by tritium and the oxygen atoms of the two free CO by the 18 O isotope. The resulting structure, shown in Figure 6 2). The Ph angle (99°) is typical of an open structure, which allows interaction between two neighbouring molecules, as it was already described for cis-ACBC. The phenyl rings are indeed involved in an interaction with the neighbouring molecule that involves a CH group of the ACBC ring (C6H••• interaction). The tetramer satisfactorily reproduces the IR absorption, as shown in Figure 5. The VCD signature is well reproduced in the region of the β(NH) bend as this region shows two bands of opposite sign. The (CO) stretch region also shows bisignate signatures, although the intensity enhancement is not reproduced in the calculations. Still, the gap between the β(NH) bend and the (CO) stretch transitions is still slightly overestimated relative to the experiment. Moreover, the ratio between the (NH) intensity to that of the (CH) is still underestimated in the calculated spectrum of the tetramer relative to the experiment. This suggests that chains larger than four units should be taken into account.

DISCUSSION: LOCALISED vs. DELOCALISED VIBRATIONAL MODES

The main difference experimentally observed between cis and trans-ACBC compounds appear in the region of the ν(CO) stretching modes. While the VCD signature in the β(NH) range is similar in shape and intensity for cis and trans-ACBC, the VCD intensity of the ν(CO) bands is one order of magnitude larger in the trans compounds. It should be noted that the intensity enhancement is also observed for the IR absorption in the region of the ν(NH) stretch. This is however not reproduced in the calculation as already described by Marty et al. in their description of VCD enhancement in perylene bisimide oligopeptides. 18 A very recent calculation of the VCD signal of solid alanine by means of ab initio molecular dynamics (AIMD) and nuclear velocity perturbation theory (NVPT) has explained the enhancement of the VCD signal in terms of supramolecular helical arrangement of the oscillators. 23 One of the important conclusions of the latter work is that the intense VCD signal of the carbonyl bands is exclusively explained in terms of supramolecular effects and that considering coupled oscillator on a minimum intermolecular distance of 4-5 Å is necessary for reproducing the experimentally observed enhancement. We can raise the question as to whether such supramolecular effects also play a role to explain the spectroscopic properties of trans-ACBC. To test this hypothesis, we have calculated a subunit of two parallel hydrogen-bonded dimers, in a geometry similar to that of two parallel strands in the crystal. However, the IR absorption of this sub-unit (not shown) is completely at odds with the experimental spectrum. These results point at a long-range organisation of the solid, along the axis of the hydrogen bonds as the main feature responsible for the observed spectral gap between (CO) and (NH) bands. Still, the coupling between two parallel strands could be responsible for intensity enhancement in the VCD spectrum of trans-ACBC. However, it is not clear why this effect would play a determining role in trans-ACBC or alanine, and not in cis-ACBC, although alanine and the two model systems for cis and trans-ACBC the three molecules have the same space group P212121. The size of the systems studied here precludes the use of AIMD methods, but polarizable force fields would be interesting for calculating the VCD spectra of such complex molecules in a solid environment. 37

Conclusion

We have studied the IR absorption and VCD spectra of Boc-(cis-ACBC)-NH-Bn and Boc-(trans-ACBC)-NH-Bn. In cis-ACBC, the fact that the VCD spectrum is well reproduced by considering the monomer as the smallest unit in the solid indicates that longdistance intermolecular couplings can be neglected. The intensity of the VCD signal of trans-ACBC is one order of magnitude larger than that of cis-ACBC in the region of the (CO) stretch. The two intense bisignate couplets observed for the (CO) stretching modes are qualitatively reproduced by a one-dimension tetramer involving the head-to-tail arrangement of the sub-units with parallel NH•••OC hydrogen bonds. Calculations on smaller model systems indicate that the gap between the (CO) stretch and the (NH) bend decreases when the chain length increases and that although the calculations have been limited to four units, including six units improves this gap and shows extensive delocalisation of the (CO) stretches in a "zigzag" fashion. The inclusion of parallel strands might be necessary for reproducing the intensity of the bisignate VCD couplets observed for the (CO) stretches but are out of reach due to the size of the systems.

Graphical Abstract

FIGURE 1

 1 FIGURE 1 Molecules under study and numbering of atoms.  = C 3 -N 4 -C 5 -C 8  = N 4 -C 5 -C 8 -C 9  = C 5 -C 8 -C 9 -N 10 Ph= C 9 -N 10 -C 11 -C 12

  crystal structure of model systems cis-ACBC* and trans-ACBC*is shown in Figure 2. Both cis-ACBC* and trans-ACBC* compounds show infinite strands of monomers bound by intermolecular NH•••OC hydrogen bonds. The chains observed for the cis derivative involve head-to-head alignment of the monomeric subunits, which results to antiparallel NH•••OC hydrogen bonds. Two sorts of hydrogen bonds are therefore expected to be spectroscopically observed, namely, N4H•••OC3 and N10H•••OC9. The crystal structure of the trans derivative shows a head-to-tail arrangement of the sub-units with parallel NH•••OC hydrogen bonds. The parallel double hydrogen bond involves N4H•••OC9 and N10H•••OC3 hydrogen bonds. If we assume a similar crystal structure for the model systems and the compounds studied here, all the phenyl rings are on the same side of the strand in the cis derivative. We can anticipate that this arrangement has consequences in terms of dispersion energy and packing effect. Conversely, the aromatic substituents are on alternating sides in the trans compound. The hydrogen bond distances and important structural parameters of the model systems are listed in Tables

FIGURE 2

 2 FIGURE 2 Crystallographic structure of a) cis-ACBC* and b) trans-ACBC* showing the head-to-head and head-to-tail arrangement of the molecules.

FIGURE 3

 3 FIGURE 3 Left: a) Experimental IR absorption of cis-ACBC (1R,2S) (red) b) calculated IR absorption of the monomer subunit Mcis c) Experimental VCD spectrum of (1R,2S) (red) d) calculated VCD spectrum of the monomer subunit (see text). Right: Experimental IR absorption of cis-ACBC (1R,2S) (red) and calculated IR absorption of e) c-1 f) c-3 g) c-2. e) Experimental VCD spectra of cis-ACBC (1R,2S) (red) and calculated VCD spectrum of h) c-1 i) c-3 j) c-2. The calculations are at the B3LYP /6-31G++(d, p) level. The harmonic frequencies are scaled by 0.97.

FIGURE 5 a

 5 FIGURE 5 a) Experimental IR absorption of (1S,2S) trans-ACBC (red). Calculated IR absorption of b) the monomer t-1 in the gas phase c) the monomer Mtrans (see text) d) the tetramer subunit. e) Experimental VCD spectrum of trans-ACBC (1S,2S) (red). Calculated VCD spectrum of f) the monomer t-1 in the gas phase g) the monomer Mtrans (see text) h) the tetramer subunit. In the tetramer; the two NH and CO of the termini have been isotopically modified by tritium and O18, respectively, to minimize boundary effects. The calculations are at the B3LYP/6-31G ++(d, p) level. The frequencies are scaled by 0.97.

  , shows the typical OH•••N double strand zigzag pattern. The intermolecular hydrogen bonds are slightly longer (d(O3•••NH10) = 1.94 Å) between the amide NH and the ester CO than between the ester NH and the amide CO (d(O9•••NH4 = 1.91 Å), as it is the case for trans-ACBC* (see Table

FIGURE 6

 6 FIGURE 6 Calculated structure of the trans-ACBC tetramer calculated at the B3LYP/6-31G(d,p) level of theory. Dashed lines indicate the hydrogen bonds.

  ATR) ν 3336, 3299, 1681, 1644, 1531; 1 H NMR (360 MHz, CDCl3) δ 1.39 (s, 9H), 1.75 (pseudo quint, 1H, 2 J = 3 J = 9.9 Hz), 1.86-1.99 (m, 1H), 2.02-2.23 (m, 2H), 2.86-2.98 (m, 1H), 4.08-4.20 (m, 1H), 4.35 (dd, 1H, 2 J = 14.8 Hz, 3 J = 5.0 Hz, ABX), 4.54 (dd, 1H, 2 J = 14.8 Hz, 3 J = 6.3 Hz, ABX), 5.14 (bd, 1H, 3 J = 6.3 Hz), 7.17-7.37 (m, 5H), 8.38 (bs, 1H, ABX);

	13 C
	NMR (90 MHz, CDCl3) δ 18.7, 24.9, 28.4, 43.3, 48.8, 50.1, 80.4,
	127.1, 127.8, 128.5, 138.8, 156.3, 173.0; HRMS (ESI+) Calcd for
	C17H24N2NaO3 [M + Na]

2 mg, 78%) as a white solid. Mp 176-177 °C; Rf 0.28 (CH2Cl2/EtOAc = 80/20); [α] D 24 -26 (c 0.50, CHCl3); IR (+ : 327.1679, found 327.1667.

  We have therefore extracted a monomer from this core, referred to as Mcis hereafter. Figure3band 3d show the calculated IR absorption and the VCD spectra calculated at the B3LYP level for Mcis. It should be noted here that optimising Mcis with dispersion correction results to the folding of the molecule and a wrong VCD signature in the region of the (CO) stretch. For this reason, we have performed the calculations at the B3LYP/6-31G++(d,p) level with no dispersion corrections. The calculations with dispersion corrections are shown in Figure

Acknowledgements

Support from the French National Research Agency (ANR) is acknowledged (Grant ANR-11-IDEX-0003-02 and ANR-17-CE29-0008).We thank Antoine Béthune for his experimental contribution. We acknowledge the use of the computing facility cluster MésoLUM of the LUMAT federation (FR LUMAT 2764).

Supporting information

Additional supporting information may be found in the online version of this article at the publisher's website.