Yang Sun

Guy André Boy

Anne Barros
email: anne.barros@centralesupelec.fr

PRODEC-based task analysis for the design of semiautomated trains

Keywords: Railway, Safety analysis, Human System Integration, PRODEC, Human-in-the-loop simulation, BPMN

The deployment of high-speed automated trains is a worldwide objective for railway stakeholders. The French railway operator SNCF has conducted several safety-oriented analyses and studies to better understand Automated Train Operation (ATO) impacts. But none of them is supported by human-the-loop simulation (HITLS). In PRODEC, AS-IS scenarios based on existing declarative configurations can help to project TO-BE scenarios based on new declarative configurations of the system to be designed. This paper is dedicated to the task analysis of the scenarios selected and constructed by safety analysis. The comparison between the train driver's tasks in AS-IS scenarios at Grade of Automation 1 (GoA1) for manual driving and the train driver's tasks in TO-BE scenarios at Grade of Automation 2 (GoA2) for teaming with ATO and ETCS enables to discover design gap and tasks evolutions to project potential functions and infrastructures.

Introduction

Automated trains take part in global increasingly autonomous mobility. This solution is expected to be more ecological, as well as to provide increased capacity and greater flexibility to the railway system. The International Association of Public Transport (UITP) and the International Electrotechnical Commission (IEC) defined four levels of automation for railway automation [START_REF] Trentesaux | The Autonomous Train[END_REF] [START_REF]Railway applications -Urban guided transport management and command/control systems -Part 1: System principles and fundamental concepts[END_REF]. From Grade of Automation 1 (GoA1) up to GoA4. Current trains operating within SNCF railway infrastructures are at GoA1 under manual driving. From SNCF automated train vision, the following decade is a key period for automated train design and deployment. The introduction of the European Rail Traffic Management System / European Train Control System (ERTMS/ETCS) and Automatic Train Operation (ATO) enables the transition from GoA1 to GoA2. At GoA2, ATO is available when its operation conditions are met. ATO can take charge of acceleration, braking following the signals and recommendations of the ERTMS/ETCS. Train driver and ATO share the driving task once ATO is activated. This paper addresses the application of PRODEC [START_REF] Boy | The machine as a partner: Human-machine teaming design using the PRODEC method[END_REF] in the railway to analyze human tasks in the early design stage of the automated train operating system on GoA2.

Human performance cannot be separated from the organization and available technology. A good understanding of the aiming technologies on automatic driving is critical for both designers and train drivers. This is the basis for the analysis of the evolution of human roles and tasks. Based on ATO and ERTMS specifications, we model the process map for selected scenarios using Business Process Model and Notation (BPMN) [START_REF]Business Process Model and Notation (BPMN)[END_REF]. BPMN standardizes the visualization of the process map and clarifies related tasks. More specifically, BPMN-formalized scenarios facilitate exchanges between various engineers and operators and support the clarification of tasks and responsibilities of the train driver when teaming with ATO. In our case, BPMN is used to support the specifications of technical systems and visualize the interaction between operators and systems. Humans differ from machines in that they do not do exactly what they are supposed to do. In other words, their activity is not necessarily the same as the task they are supposed to perform. Activity results from the performance of task executed by cognitive and/or physical functions [START_REF] Boy | Cognitive Function Analysis[END_REF]. Environment and allocatable resources can also affect the results of task execution. PRODEC is a scenario-based design method [START_REF] Carroll | Scenario-Based Design[END_REF], [START_REF] Boy | Design for Flexibility: A Human Systems Integration Approach[END_REF], based on two fundamental concepts: the declarative configurations of the system and the procedural scenarios [START_REF] Boy | The machine as a partner: Human-machine teaming design using the PRODEC method[END_REF], [START_REF] Boy | PRODEC for Human Systems Integration of Increasingly Autonomous Systems[END_REF]. Since human activity is complex and hard to predict, human-in-the-loop simulation (HITLS) allows us to observe the natural human reaction in a simulated environment close enough to reality. By observing the differences between tasks from designed scenarios and activities from simulation results, emergent functions can be elicited and contribute to improve our initial design. More generally, these comparisons enable the discovery of emergent properties, projecting new functions and potential infrastructure(s) that will define the projection of the next TO-BE configurations of the sociotechnical systems being designed and developed. This human-centered approach is based on safety, efficiency, and comfort principles that need to be defined.

2

Automated trains and safety-oriented PRODEC application

From manual driving to autonomous trains: the different grades of automation in railway

Automated trains can provide the modern railway system a better fluidity at a lower energy cost [START_REF] Wang | Survey on Driverless Train Operation for Urban Rail Transit Systems[END_REF]. The International Electrotechnical Commission (IEC) defined the requirements and functions for four grades of automation in railway applications [START_REF]Railway applications -Urban guided transport management and command/control systems -Part 1: System principles and fundamental concepts[END_REF] from GoA1 to GoA4. GoA1 represents the fully manual-operated trains with Automatic Train Protection (ATP). Most of the long-distance trains today work under this automation level. Telecommunication and information technology maturity increasing, ATO is now available to be integrated in the driving cabin on GoA2. At GoA2, ATP becomes a part of ERTMS by following the journey profile and signalizations. Nevertheless, at this level of automation, the driver is always present in the driving cabin. The driver oversees the doors open/close functions and assumes responsibility for managing the vehicle in the event of an ATO dysfunction. On GoA3, the whole train driving cabin does not exist anymore. A well-trained agent is supposed to be on board with the passengers to supervise train operation and take charge in case of emergency. And finally, autonomous trains on GoA4 are supposed to be unattended and fully automated. At this level, no agent is required to stay onboard. This level requires great capacity for situation awareness and decision making from the overall sociotechnical system. Emphasizing the challenge today, from GoA1 to GoA2, the key evolution for train drivers is the introduction of ATO. When operating conditions are met, the ATO indicator illuminates to inform the train driver that the ATO is ready. Train driver can decide to either engage or disengage ATO during the journey. Once activated, the ATO can take over of the acceleration/braking according to the information provided by the ETCS. When the ATO is not engaged, manual driving is still possible, and the driving condition goes back to the same at GoA1. Regardless of these changes, at GoA1 and GoA2, the train driver always takes full responsibility for the safety and security of train operation under either situation.

From a system view, a system can represent either a natural or an artificial entity [START_REF] Boy | Tangible Interactive Systems: Grasping the Real World with Computers[END_REF]. It means it can be either a human or a machine, hardware or software, concrete or abstract. A system can be cognitive, physical, or both [START_REF] Boy | Human-centered design of complex systems: An experience-based approach[END_REF]. Research results showing the introduction of automation in railway operation underloads the train driver from highly stressed work [START_REF] Brandenburger | Automation in Railway Operations: Effects on Signaller and Train Driver Workload[END_REF]. With the train driver's task reduction, we need to reassess the related human factors. This reassessment includes cognitive workload, attention level, and situation awareness, all factors that affect the performance of the train drivers' cognitive functions. The performance of these cognitive functions are hard to evaluate. A detailed analysis for tangible tasks performance enables the evaluation of related cognitive functions' performance by observing the differences between activities and tasks.

To identify the gap between AS-IS and TO-BE scenarios, we must first get an idea of the specifications and requirements of the future system to be designed. Back in the early 2000s, various railway signaling systems exist in European Union's member states. For example, the Thalys train which crosses the border of French, Germany, Belgium, and Netherlands is embedded with seven types of signaling systems. The European Rail Traffic Management System (ERTMS) is under deployment to increase interoperability in the EU. ERTMS includes the Global System for Mobile communications for Railways (GSM-R), which provides communication between the train and the Radio Block Center (RBC), and the European Train Control System (ETCS). ETCS provides Automatic Train Protection (ATP) and cabin signaling functionality. The cabin signaling calculates a target advice speed for each segment of the journey and displays this information to the train driver through the Driver Machine Interface (DMI) in the driving cabin. Under manual driving, the train driver decides when to accelerate or brake according to this information and the journey profile he receives before the journey.

Fig. 1.

ERTMS system architecture at GoA1 (adapted from [START_REF] Era | ERTMS/ATO. System Requirements Specification[END_REF]).

In the GoA2 configuration, trains must deploy the ATO. The ATO consists of two parts: ATO-Onboard (OB) and ATO-Trackside (TS). ATO-OB is onboard the trains and automatically drives the train while it is engaged. Before the journey starts, ATO-TS sends the Journey Profile (JP) to the ATO-OB. It contains information about the specific train (train length, brake percentage, brake position), the rail track profile (e.g., uphill, downhill, gradient, track profile), the scheduled timetable for arriving at a train station, and the ATO Operational Speed Profile. Using the JP, ATO can operate the trains based on the ETCS cabin signaling advice. Figure 2 shows the reference architecture of ERTMS/ATO. ATO frees train drivers from speed control of the trains between train stations. Nevertheless, the train driver is still in charge of door operations and communications between the other entities. Fig. 2. ERTMS/ATO system architecture at GoA2 (adapted from [START_REF] Era | ERTMS/ATO. System Requirements Specification[END_REF]). This teaming between the train driver and ATO is a concrete example of Human-Machine Teaming (HMT). HMT consolidates human-centered design and systems engineering to achieve Human System Integration (HSI) by combining human and machine properties and attributes in the same package. Indeed, a system can include machines and people, and people can include machines [START_REF] Boy | The machine as a partner: Human-machine teaming design using the PRODEC method[END_REF]. Appropriate physical and/or cognitive functions must then be allocated to the right human and machine systems within the overall system of systems. This is the purpose of the PRODEC method that starts by developing task analyses to better identify emerging functions to be further allocated.

Scenarios construction for safety-oriented PRODEC application

Following a good understanding of the specifications and needs for the future semiautomated trains at GoA2, we need to choose and construct the typical scenarios during train driving to build an effective presentation of train operations. Safety is always the highest priority in train operation. From our earlier research that included accident analysis and experts interviews, we identified several safety-critical conditions and events for railway operation: signaling, rail conditions, and one safety-critical area: the station [START_REF] Sun | Preliminary Safety-Critical Scenario Analysis of Semi-Automated Train Operation[END_REF]. For manual driving trains, several types and simulators are available for training purpose within the SNCF. From these existing training scenarios and the declarative knowledge that we learnt from practice; we selected five typical AS-IS scenarios. Based on the specifications and requirements of the future ATO application at GoA2, we foresee 5 corresponding TO-BE scenarios.

One scenario can be represented by both on-board and environmental components. Suitable environments are as the train itself in ensuring the full functionality of the rail system. For environmental factors, an obstacle on the rail can simply turns a nominal situation to an emergency on both GoA1 and GoA2. The train driver remains responsible for monitoring the environment, regardless of the degree of automation. Regarding on-board components, the safety analysis shows that signaling is another essential element for safe train operation. The most frequent type of incident in the last 8 years on SNCF network is: inadvertent crossing of a closed signal. 91% of this type of error is related to driver error according to the Rail Safety and Standards Board (RSSB) report [START_REF]How to use the SPAD toolbox[END_REF]. We also interviewed a group of train drivers for their opinions on the safety analysis. From their point of view, one critical area during the journey is entering train station. The environment perception, signalization reading, and speed control are the difficulties for entering a station and stopping the train precisely on the platform. The weather is an important factor for this operation. Rain, wind and dead leaves fell on the rail can highly impact train drivers' judgement to stop the train. Therefore, we add the weather factor into our scenarios for the enter in destination station driving phase.

We Human-in-the-loop simulation (HITLS) allows to elicit train drivers' behaviors in these different driving situations. Before practicing the simulation, several preparatory works are necessary. We first divide the train journey into several driving phases: leaving the departure station, entering the high-speed zone, driving in the high-speed zone, leaving the high-speed zone, entering the destination station. A task analysis is performed for each driving phase for AS-IS scenarios (GoA1) and TO-BE scenarios (GoA2). Then we perform physical and cognitive analysis with AUTOS pyramid [START_REF] Boy | Cognitive Function Analysis[END_REF]. Human and machines share the operation mission by performing series of functions. To ensure the train safety, some human tasks and machine tasks can be the same. For instance, from the specifications for GoA2 operation, the manual driving is always at train driver's disposal. This is a safety barrier for train operation. One possible situation is: ATO disengages without train driver's order. Under this emergency & near accident situation, the train driver needs to quickly get awareness of the situation and retake the task of acceleration/ brake according to the signalization.

3

Task analysis supports declarative modeling.

BPMN enables the visualization of scenarios and clarification of exchanges between various entities within the operating system. Each entity has his own lane in BPMN.

The entity can be either human or machine. One technical system can be divided into several subsystems to precise his function. In practice, operators are not always familiar with modeling technologies and stakeholders' expectations. This visualized process model is easier to understand by all entities in system engineering. Human operators of different roles can identify themselves in BPMN to clarify his/her tasks and interactions with other entities before practice. By involving different entities and roles, we can align the stakeholders' expectations and the domain experts' fields experience in one common agreed model. As a step of PRODEC application, we modeled manual driving (AS-IS) scenario and semi-automated driving (TO-BE) scenario in normal situations. We did a task analysis for the two automation grades to analyze the impact of the introduction of ATO on train operation. Task analyses clearly supported the elicitation of emergent properties that contributed to the definition of appropriate (declarative) functions.

AS-IS model and function analysis for typical & normal situation

A typical situation model on GoA1 enables a task analysis of the daily tasks and functions performed by different entities in railway operation. Before tasks execution, each role in the operational system should be aware of his tasks and functions to perform. Figure 4 presents a AS-IS typical & normal scenario at GoA1. Three roles present in three lanes: train driver, signaler, and other agents. One journey cannot be completed successfully without a harmony cooperation between these roles. From this BPMN model, we summarize and define the following functions for these three roles (Table1). A preliminary work of definition of the various functions enables the association of each task with the right function [START_REF] Boy | PRODEC for Human Systems Integration of Increasingly Autonomous Systems[END_REF]. Following Endsley and Rasmussen models [START_REF] Rasmussen | Skills, rules, and knowledge; signals, signs, and symbols, and other distinctions in human performance models[END_REF], we categorized these functions in three types: Situation Awareness (SA), Decision-Making(DM), and Action Taking(AT).

AT

In a typical & normal situation, the involved operational roles are supposed to be capable of performing the train operation by following the procedures. So, in this situation, most functions are action taking and situational awareness. Anomalies can be observed by inspections and verifications. The normal situation then can change into critical or near accident situations.

TO-BE model and function analysis for typical & normal situation

Based on the AS-IS model and ATO over ETCS specifications, we can project to the semi-automated trains (TO-BE) operation scenarios at GoA2. Fig. 5. is BPMN model for a typical & normal scenario. At this automation grade, train driver and other agents are always presented in the model. There are two important changes on the roles in operating system: i) ATO is embedded for train operation. ii) The signaler's role is replaced by ERTMS. Therefore, we find four lanes in this BPMN model. Same work of preliminary function definition is done for this scenario. At GoA2, the ATO and ETCS functions are analyzed for the first time. ETCS takes charge of the optimal speed calculation following the signalizations and speed limit. ETCS then sends this information to ATO for acceleration/brake control. ATO has two principal functions: status indication and drive when engaged. When ATO operational conditions are met, the ATO informs the train driver that it's ready to be engaged. The engagement/disengagement of ATO is decided by train driver. When ATO is ready, the train driver can decide to engage ATO for automated driving. ATO disengages when its operation conditions are no longer met. However, during the journey where ATO operation conditions are met, train driver can still decide to disengage ATO if he finds manual driving is more suitable for the current journey segment. This decision of when the train driver engage/disengage ATO is a question of trust and collaboration between human operators and automated machines [START_REF] Lee | Trust in automation: Designing for appropriate reliance[END_REF].

Depending on the situation awareness and train driver's field experience, he/ she can have different trust levels to ATO under different situations. And these different trust levels can lead to different decisions.

AT

Once ATO is engaged, the train driver's workload is significantly reduced. Compared to AS-IS scenario, the train drivers' principal tasks are supervision and perception. Along with this advantage, another question of how to keep train drivers' attention is raised. In current driving cabins, the automatic vigilance system requires the train driver to press a pedal every 30 seconds to confirm train driver's presence and vigilance. From research in aviation, a shared responsibility of human machine teaming can keep pilots involved and improve pilots performance and situational awareness [START_REF] Rogers | Automation Promises and Concerns for Three Levels of Automation: A Survey of Pilots of Advanced Automation Commercial Aircraft[END_REF]. This next step of human machine teaming in next generation of semiautomation trains can potentially be an optimal solution for train driver's vigilance.

Analysis between AS-IS and TO-BE

One new function for train driver at GoA2 is the decision making to engage/disengage ATO. Such decision is impacted by operational context and the allocatable resources for train driving. During different driving phase, the train driver can make different decision to engage/ disengage ATO. To evaluate the trust and collaboration between train driver and ATO, we propose to observe the following vigilant points during the Human-in-the-loop simulation: i) time interval from ATO ready to train driver's engagement of ATO. ii) duration of ATO engaged iii) when the train driver disengages ATO. After ATO indicates its 'ready' status, the time interval until train driver engages ATO is an indicator for trust between train driver and ATO.

Trust can be seen at different levels. To start with, the train driver needs to know what ATO is capable of. From ATO over ETCS specifications, ATO can strictly follow ETCS to achieve best performance for train journey. Then on next level, train driver evaluates the quality of feedback information provided by ATO. The train driver evaluates ATO performances by supervising the signals and the speed limits.

Based on the task analysis, the eye tracking and ECG (electrocardiogram) equipment are to be applied during human-in-the-loop simulation. By analyzing human behaviors by appropriate algorithms, we can evaluate human cognitive loads for each task and functions associated. This analysis also enables us to evaluate train drivers' technical and non-technical knowledge for train driving. Once a weak point identified, we can either reinforcement the training for concerning knowledge or provide more information to facilitate train drivers' operation.

4

Conclusions and perspectives.

This paper is a part of the preliminary work 1 for the application of the PRODEC method. Before practicing HITLS, we identified the tasks for each role in railway operation following the specifications of ATO/ETCS and field experience. We compared AS-IS and TO-BE scenarios to identify the task changes from GoA1 to GoA2. After this preliminary work, the next step will be the HITLS on GoA1 and GoA2 simulators to observe the difference between tasks and activities.

According to the current specifications, the ATO over ETCS is not yet sufficient to allow the deployment of fully autonomous trains on the GoA3 or GoA4. Our studies have shown that machines must support situational awareness and decision making for the application of autonomous trains in GoA3/4. Artificial intelligence (AI) is a promising solution to make optimal decisions for train operations.

  classify the driving situations in three types: Typical & normal situation, Critical & Abnormal situation, and Emergency & Near accident situation. Human errors can be committed at all three kinds of situations. Minor infractions of procedures are often related to insufficient cognitive resources for physical and/or cognitive functions. Technical failures of life-critical systems can lead to different kind of driving situations depending on the driving phase of the train. Taking example of a highspeed train, dysfunction of signalization screen display before entering high-speed zone lead a typical & normal situation to a critical & abnormal situation. Because in this phase, the train driver can refer to the track side signalization to continue the train operation. However, in high-speed zone, the same technical failure leads a typical & normal situation directly to an emergency & near accident situation.

Fig. 3 .

 3 Fig. 3. Situation transitions in different phases.

Fig. 4 .

 4 Fig. 4. AS-IS model under BPMN for typical & normal situation.

Fig. 5 .

 5 Fig. 5. TO-BE model under BPMN for typical & normal situation.

Table 1 .

 1 Function analysis for AS-IS scenario in typical & normal situation .

	Role	Function	Definition	Type
	Train driver	Technical check	Check the technical systems before start-	SA
			ing the train
	Train driver	Observe signal-	Observe the signals, speed limit, and	SA
		ization	journey profile
	Train driver	Environment	Observe the operational environment	SA
		perception
	Train driver	Drive	Accelerate/ Brake following the signaliza-	AT
			tions
	Train driver	Press the auto-	Press the automatic vigilance system	AT
		matic vigilance	pedal every 30 seconds to confirm the
		system pedal	driver's level of alertness
	Signaler	Signal control	Control signals according to the railway	AT
			traffic and journey profile
	Other agents	Safety control	Ensure the onboarding/ offboarding of	SA
			the passengers and no luggage is left
			behind
	Other agents	Send authoriza-	Send authorization for train departure to
		tion	the train driver

Table 2 .

 2 Function analysis for TO-BE scenario in typical & normal situation.

	Role	Function	Definition	Type
	Train driver	Technical check	Check the technical systems before start-	SA
			ing the train
	Train driver	Observe ETCS	Observe ETCS status	SA
		status
	Train driver	Observe ATO	Observe ATO status	SA
		status
	Train driver	Engage / Disen-	Decide to engage /disengage ATO when	DM
		gage ATO	the ATO status is 'ready to engage'
	Train driver	Environment	Observe the operational environment	SA
		perception
	Train driver	Press the auto-	Press the automatic vigilance system	AT
		matic vigilance	pedal every 30 seconds to confirm the
		system pedal	driver's level of alertness
	ETCS	Journey profile	Calculate the optimal speed for current	AT
		computation	segment and send this information to
			ATO
	ETCS	Signal control	Control signals according to the railway	AT
			traffic and journey profile
	ATO	Drive	Accelerate/ Brake following ETCS	AT
	ATO	Status indica-	Inform train driver the current ATO status AT
		tion
	Other agents	Safety control	Ensure the onboarding/ offboarding of	SA
			the passengers and no luggage is left
			behind
	Other agents	Send authoriza-	Send authorization for train departure to
		tion	the train driver

Therefore, this contribution is a work in progress in a doctoral program that will continue to be published in another article to come. It was important to publish this account to describe the method and first task analysis results.


Zhehong Gan
email: gan@magnet.fsu.edu

Ivan Hung

Yusuke Nishiyama

Jean-Paul Amoureux

Olivier Lafon

Hiroki Nagashima

Julien Trebosc

Bingwen Hu

Julien Trébosc

N-14 overtone nuclear magnetic resonance of rotating solids

Keywords: 14 N, overtone NMR, double-quantum transition, magic-angle spinning, quadrupole spin, Floquet theory, second-order quadrupular shift, spinning induced shift, serious-connected hybrid magnet

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Nuclear magnetic resonance (NMR) is a powerful tool for characterizing structure and dynamics of polycrystalline and disordered solids. Much of its success relies on its capability to resolve the chemical environments of atomic sites through slight variations of their resonance frequency. Nuclear spin interactions, such as chemical shift and dipolar or quadrupolar couplings, are anisotropic in nature and thus their orientation dependence leads to line broadening for powder samples. The anisotropic broadening must be averaged or reduced to achieve high spectral resolution. In solutions, the averaging occurs naturally due to rapid isotropic molecular tumbling.

In solids, magic-angle spinning (MAS) of the sample averages out rank l = 2 spin interactions like dipolar coupling and chemical shift anisotropy (CSA). 1,2 For nuclear spin values larger than 1/2, the quest for high spectral resolution is confronted by the much larger interaction between the electric field gradient (EFG) and the electric quadrupole moment of the nucleus. The magnitude of the quadrupole interaction is often in the megahertz range, far larger than the fastest sample spinning frequency available. Fortunately, the first-order quadrupole interaction vanishes for the +1/2 ↔ -1/2 central transition of half-integer spins, allowing for its spectral acquisition without the large first-order broadening. The remaining second-order interaction is much smaller and can be partially reduced with MAS. Innovative methods like double rotation (DOR), 3 dynamic angle spinning (DAS), 4,5 multiple-quantum and satellite transition MAS (MQMAS and STMAS), 6,7 have been developed for the complete removal of the second-order quadrupolar broadening that has angular dependence up to rank l = 4. High magnetic fields can also reduce directly the secondorder broadening, making solid-state NMR of quadrupolar nuclei one of the most important driving forces for high-field NMR. [8][9][10] The capability to obtain high spectral resolution has indeed facilitated the widespread use of solid-state NMR for quadrupolar nuclei, which constitute the majority of isotopes in the Periodic Table. 11,12 14 N is the most abundant isotope (99.65%) of nitrogen, an important element for all branches of chemistry, and one of the few nuclei in the Periodic Table with an integer spin. It is a spin S = 1 nucleus with a moderate quadrupole moment (20.44 × 10 -31 m 2 ). 13 The direct NMR detection of 14 N nuclei in solids is challenging as it often requires specialized experimental approaches in order to excite and observe spectra that are typically several MHz wide (e.g. broadband MAS, field-or frequency-stepped piecewise acquisition, broadband frequency sweep pulses, etc.). Obtaining site resolution is even more difficult for solids containing distinct 14 N sites. For samples with 13 C or 1 H nuclei near nitrogen sites, two-dimensional (2D) MAS experiments, like heteronuclear multiple-quantum correlation (HMQC), have been introduced to observe 14 N nuclei indirectly, partially overcoming the resolution and sensitivity difficulties of direct observation. [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54] In this work, we focus on an approach, called nitrogen-14 overtone ( 14 N OT ) NMR, which directly excites and acquires 14 N spectra at twice its Larmor frequency. The m = +1 ↔ -1 double-quantum (DQ) transition is usually considered as forbidden, but it can become directly observable in the presence of large quadrupole interactions. The main advantage of 14 N OT NMR is that the first-order quadrupolar broadening vanishes similarly to the central transition of half-integer nuclei. 14 N OT NMR was first demonstrated experimentally by LeGros and Bloom, 55,56 and later applied to biomolecules. 57-60 14 N OT spectra of rotating samples have been recorded by Tycko and Opella, aiming at further line narrowing of CSA and second-order quadrupolar broadenings. However, for the sample selected and the low magnetic field and spinning frequency used, the spectra were found to be complicated by overlapping spinning sidebands. As a result, no significant line narrowing was materialized by sample rotation at that time. 57 On the theoretical aspect, 14 N OT NMR is different from conventional NMR with single-quantum excitation and detection. Tycko and Opella developed a formalism based on perturbation theory. 57 Two later works by Marinelli et al. 61 and Trease et al., 62 provided more general descriptions in the form of density operators. They also explored the possibilities of applying advanced methods like DOR and DAS to completely average the quadrupolar broadening of 14 N OT spectra. Numerical simulations were performed, which indicated that possible complications may occur if those methods, which were developed originally for half-integer quadrupolar nuclei, were applied to 14 N OT . However, no experimental demonstration was performed.

Recently, O'Dell and Ratcliffe published an experimental study of 14 N OT NMR under MAS using higher magnetic fields, faster spinning rates and samples with smaller quadrupolar couplings. 63 Simple 14 N OT MAS spectra were obtained with line shapes typical of second-order quadrupolar patterns, showing potential for measuring 14 N chemical shifts and quadrupolar coupling parameters. One of their main findings is that the 14 N OT peak position shifts under MAS by twice the spinning frequency (ωr) with respect to the non-spinning case. A brute-force simulation has confirmed this shift and predicted that its sign depends on the relative sense of spinning with respect to the magnetic field. 64 However, no theory or explanation was given for this intriguing feature. Recently, overtone excitation and detection have also been combined with double rotation, and the results showed that DOR can completely cancel the second-order quadrupolar broadening of 14 N OT spectra. 65 All these experimental observations have renewed our interest in reexamining the 14 N OT NMR theory, particularly under sample rotation.

In this article, we analyze the results observed in one-dimensional pulse-and-acquire 14 N OT experiment. First, the 14 N OT NMR theory for the time independent case of static samples is presented, using a density operator formalism similar to the work of Marinelli et al. 61 and Trease et al.. 62 An overtone parameter, , is introduced such that 14 N OT NMR can be described in a similar way as NMR applied to spin S = 1/2 nuclei. In the case of rotating samples, Floquet theory is used to treat this time dependent problem to obtain analytical expressions for the excitation and detection of the 14 N OT transition. With spinning samples, the overtone parameter becomes time dependent and consists of five components ξk (k = 0, 1 and 2) modulated at 0, r and 2r, respectively. The relative amplitudes of these terms reveal that under MAS rotation the dominant component is either the +2r or -2r sideband, depending on the spinning direction relative to the magnetic field B0, making the main overtone peak shift with the spinning frequency. In addition to the explanation of the intriguing MAS overtone features, the theory can also be used for rapid numerical simulations of overtone excitation and spectral line shapes.

Theory for static samples

In the laboratory frame

Let us consider a spin S = 1 nucleus in the laboratory frame (L) where the magnetic field is along the z axis. The Hamiltonian including the Zeeman, quadrupole and rf interactions in this frame can be written as: 66 are the Larmor and quadrupolar coupling frequencies, respectively, where eQ is the electric quadrupole moment of the S spin and eq is the principal component of the EFG at the position of the S nucleus. The linearly modulated rf field, produced by the excitation coil tilted in the xz-plane at an angle θC with respect to the magnetic field B0, is described by its irradiation frequency irr, phase , and nutation frequency 21 = 2B1, where B1 is the peak amplitude of the rf field. Fig. 1

𝐻 𝐿 = 𝐻 𝑆 𝐿 + 𝐻 𝑟𝑓 𝐿 (1)
𝐻 𝑆 𝐿 = 𝜔 0 𝑆 𝑧 + 𝐻 𝑄 𝐿 ( 2
)
𝐻 𝑟𝑓 𝐿 = 2𝜔 1 (𝑆 𝑧 cos 𝜃 𝐶 + 𝑆 𝑥 sin 𝜃 𝐶 ) cos(𝜔 𝑖𝑟𝑟 𝑡 + 𝜙) (3)
𝐻 𝑄 𝐿 = 𝜔 𝑄 ∑ (-1) 𝑚 𝐴
where (PL, PL, PL) are the Euler angles defining the orientation of the P frame in the L frame.

The 𝐴 2,𝑚 ′ 𝑄,𝑃 components are given by 𝐴 2,0 𝑄,𝑃 = √6, 𝐴 2,±1 𝑄,𝑃 = 0, 𝐴 2,±2 𝑄,𝑃 = -𝜂 𝑄 /2 (6) where ηQ is the asymmetry parameter of the EFG tensor. The elements of the Wigner matrix have the following form

𝐷 𝑚 ′ 𝑚 2 (𝛼 𝑃𝐿 , 𝛽 𝑃𝐿 , 𝛾 𝑃𝐿 ) = exp(-𝑖𝑚 ′ 𝛼 𝑃𝐿 ) 𝑑 𝑚 ′ 𝑚 2 (𝛽 𝑃𝐿 ) exp(-𝑖𝑚𝛾 𝑃𝐿 ) (7)
where 𝑑 𝑚 ′ 𝑚 2 (𝛽 𝑃𝐿 ) are the reduced Wigner matrix elements. It can be shown that the complex conjugate of 𝐴 2,𝑚 𝑄,𝐿 is equal to

(𝐴 2,𝑚 𝑄,𝐿 ) * = (-1) 𝑚 𝐴 2,-𝑚 𝑄,𝐿 (8)
The spin irreducible spherical tensor operators for the quadrupolar coupling are given by

𝑇 2,0 𝑄 = 1 √6 [3𝑆 𝑧 2 -𝑆(𝑆 + 1)𝟏] 𝑇 2,±1 𝑄 = ∓ 1 2 [𝑆 ± 𝑆 𝑧 + 𝑆 𝑧 𝑆 ± ] (9)
𝑇 2,±2 𝑄 = 1 2 𝑆 ± 𝑆 ±
where 1 denotes the unity operator. These tensor operators satisfy the equation

𝑇 2,𝑚 𝑄, † = (-1) 𝑚 𝑇 2,-𝑚 𝑄 ( 10
)
where 𝑇 2,𝑚 𝑄, † denotes the adjoint operator of 𝑇 2,𝑚 𝑄 .

In the L frame, the evolution of the density operator  L (t) from its initial state  L (0) is governed by the Liouville-von Neumann equation

𝑑𝜎 𝐿 (𝑡) 𝑑𝑡 = -𝑖[𝐻 𝐿 , 𝜎 𝐿 (𝑡)] (11)
The expectation value of any observable represented by an operator O L in the L frame, such as the NMR signal s(t), is given by

𝑠(𝑡) = 𝑇𝑟[𝑂 𝐿 † 𝜎 𝐿 (𝑡)] (12)
Using the same coil tilted at the angle θC for detection, the 14 N OT signal also contains longitudinal and transverse components similar to the rf irradiation, and hence the detection operator 𝑂 𝑑𝑒𝑡 𝐿 can be written in the laboratory frame as

𝑂 𝑑𝑒𝑡 𝐿 = 𝑆 𝑧 cos𝜃 𝐶 + 𝑆 𝑥 sin𝜃 𝐶 (13)
Fig. 1. Depiction of the B1 rf field generated in the xz-plane by a rf solenoid coil at an angle θC with respect to the main magnetic field B0 as typically used in magic-angle spinning probes.

In the diagonal frame

Diagonal transformation of the spin Hamiltonian

We first transform the 𝐻 𝑆 𝐿 Hamiltonian into the diagonal frame, D, where it is represented by a diagonal matrix, 𝐻 𝑆 𝐷 , with elements that represent the energy levels of the S spin,

𝐻 𝑆 𝐷 = 𝑇 -1 𝐻 𝑆 𝐿 𝑇, ( 14
)
where T is a unitary matrix describing the transformation between the D and L frames, and

T -1 =
T † is the inverse of the matrix T. In the absence of rf irradiation, the evolution operator or propagator, exp(-𝑖𝐻 𝑆 𝐷 𝑡), can be described simply by the energy levels. Similarly, the density and detection operators in the D frame can be expressed by the same transformation from those in the L frame as

𝜎 𝐷 (𝑡) = 𝑇 -1 𝜎 𝐿 (𝑡)𝑇 𝑂 𝑑𝑒𝑡 𝐷 = 𝑇 -1 𝑂 𝑑𝑒𝑡 𝐿 T. ( 15
)
Free evolution of the NMR signal can then be calculated in the D frame as:

𝑠(𝑡) = Tr[𝑂 𝑑𝑒𝑡 𝐷, † exp(-𝑖𝐻 𝑆 𝐷 𝑡) 𝜎 𝐷 (0) exp(𝑖𝐻 𝑆 𝐷 𝑡)] = Tr[𝑇 -1 𝑂 𝑑𝑒𝑡 𝐿, † 𝑇 • exp(-𝑖𝐻 𝑆 𝐷 𝑡) 𝑇 -1 𝜎 𝐿 (0)𝑇 exp(𝑖𝐻 𝑆 𝐷 𝑡)] (16)
For S = 1/2 nuclei, this transformation is usually neglected because the perturbations from the chemical shifts, scalar and dipolar interactions to the spin states quantized by the Zeeman interaction are very small compared with the much larger Larmor frequency. Thus, the energy levels of the 𝐻 𝑆 𝐷 matrix can then be obtained by simply discarding the small off-diagonal terms of all spin interactions that do not commute with the Zeeman Hamiltonian. Under this 'secular' approximation the NMR signal excitation and detection operators, usually a linear combination of Sx, Sy and Sz, have non-zero elements connecting only the single-quantum transitions. Therefore, multiple-quantum or overtone transitions are 'forbidden', which means that they cannot be directly excited and detected. It must be mentioned that in the form of multi-dimensional experiments, multiple-quantum transitions can be easily detected indirectly via spin coherence transfer through single-quantum signal observation 42,44,49,53,[START_REF] Cavadini | [END_REF]68 including the overtone transition of 10 B with S = 3 reported recently.

The diagonal transformation T is the key point for 14 N OT NMR, as it makes direct excitation and detection of the forbidden double-quantum (DQ) transition possible. Indeed, in the case of large quadrupole interactions, 𝐻 𝑄 𝐿 contains sizable off-diagonal non-secular elements that can be written for S = 1 nuclei (e.g., 14 N) as:

𝐻 𝑄 𝐿 -𝜔 𝑄 𝐴 2,0 𝑄,𝐿 𝑇 2,0 𝑄 = 𝜔 𝑄 [ 0 𝐴 2,-1 𝑄,𝐿 √2 ⁄ 𝐴 2,-2 𝑄,𝐿 -𝐴 2,1 𝑄,𝐿 √2 ⁄ 0 -𝐴 2,-1 𝑄,𝐿 √2 ⁄ 𝐴 2,2 𝑄,𝐿 𝐴 2,1 𝑄,𝐿 √2 ⁄ 0 ] (17)
Assuming ωQ < ω0, the derivation of the transformation matrix T and the resulting energy levels of matrix 𝐻 𝑆 𝐷 can be treated to first-order with static perturbation theory in operator form 71 :

𝑇 ≈ 𝟏 + 𝜀𝑉 with 𝑉 = ∑ (-1) 𝑚 𝐴 2,𝑚 𝑄,𝐿 𝑇 2,-𝑚 𝑄 /𝑚 𝑚=±1,±2 (18)
where

𝜀 = 𝜔 𝑄 𝜔 0 ⁄ (19)
is the ratio between the quadrupole and Zeeman interactions. Using T -1 = T † and Eqs. 8, 10 and 18, it can be shown that

𝑇 -1 ≈ 𝟏 -𝜀𝑉 (20)

The internal spin Hamiltonian in the diagonal frame

Using perturbation theory under static conditions and Eqs. 14, 18 and 20, we can derive the internal spin Hamiltonian in the diagonal frame

𝐻 𝑆 𝐷 ≈ (𝟏 -𝜀𝑉)𝐻 𝑆 𝐿 (𝟏 + 𝜀𝑉) = 𝐻 𝑆 𝐿 + 𝜀[𝐻 𝑆 𝐿 , 𝑉] -𝜀 2 𝑉𝐻 𝑆 𝐿 𝑉 (21)
By neglecting the highest-order term 𝜀 2 𝑉𝐻 𝑆 𝐿 𝑉 and substituting 𝐻 𝑆 𝐿 by Eq. 2, the above expression can be written as

𝐻 𝑆 𝐷 ≈ 𝜔 0 𝑆 𝑧 + 𝐻 𝑄 𝐿 + 𝜀[𝜔 0 𝑆 𝑧 , 𝑉] + 𝜀[𝐻 𝑄 𝐿 , 𝑉] (22)
By substituting 𝐻 𝑄 𝐿 by Eq. 4 and V by Eq. 18, we obtain

𝐻 𝑆 𝐷 ≈ 𝜔 0 𝑆 𝑧 + 𝜔 𝑄 𝐴 2,0 𝑄,𝐿 𝑇 2,0 𝑄 + 𝜔 𝑄 2 𝜔 0 ∑ ∑ (-1) 𝑚 𝑚 𝐴 2,𝑚 ′ 𝑄,𝐿 𝐴 2,𝑚 𝑄,𝐿 [𝑇 2,𝑚 ′ 𝑄 , 𝑇 2,𝑚 𝑄 ] 𝑚=±1,±2 2 𝑚 ′ =-2 (23)
≈ 𝜔 0 𝑆 𝑧 + 𝐻 𝑄 𝐷,(1) + 𝐻 𝑄 𝐷,(2) (24)
The third term in Eq. 22 cancels the off-diagonal elements (m ≠ 0) of the second term for the diagonalization 𝐻 𝑄 𝐿 because of the commutation relation

[𝑆 𝑧 , 𝑇 2,-𝑚 𝑄 ] = -𝑚𝑇 2,-𝑚 𝑄 72 . 𝐻 𝑄 𝐷,(𝑛) , n = 1
and 2, denote the first-and second-order quadrupole Hamiltonians in the D frame, respectively.

We can disregard the off-diagonal elements of the third term in Eq. 23 and keep only the diagonal elements for 𝐻 𝑄 𝐷,(2)

𝐻 𝑄 𝐷,(2) = 𝜔 𝑄 2 𝜔 0 ∑ 𝐴 2,𝑚 𝑄,𝐿 𝐴 2,-𝑚 𝑄,𝐿 [𝑇 2,-𝑚 𝑄 , 𝑇 2,𝑚 𝑄 ]/𝑚 2 𝑚=1 (25)
which for an isolated spin S = 1 nucleus is equal to

𝐻 𝑄 𝐷,(2) = 𝜔 𝑄 2 𝜔 0 (-𝐴 2,1 𝑄,𝐿 𝐴 2,-1 𝑄,𝐿 + 𝐴 2,2 𝑄,𝐿 𝐴 2,-2 𝑄,𝐿 )𝑆 𝑧 = 𝜔 𝑄 (2) 𝑆 𝑧 (26)
The first-order quadrupole interaction, 𝜔 𝑄 𝐴 2,0 𝑄,𝐿 𝑇 2,0 𝑄 , vanishes for the DQ transition between energy levels +1 ↔ -1, and hence the Hamiltonian in the absence of rf irradiation describing the energy levels and transition frequencies for 14 N OT NMR is given by 𝐻 𝑆 𝐷𝑄 = 2(𝜔 0 + 𝜔 𝑄 (2) )𝑆 𝑧 𝐷𝑄 = 𝜔 𝐷𝑄 𝑆 𝑧 𝐷𝑄 (27) where 𝜔 𝑄 (2) is the second-order quadrupole frequency. Here we have used a single-transition spin S = 1/2 operator 𝑆 𝑧 𝐷𝑄 to describe the +1 ↔ -1 transition. [START_REF] Ernst | Principles of Nuclear Magnetic Resonance in One and Two Dimensions[END_REF] The factor of 2 in Eq. 27 comes from the reduction from the spin S = 1 Sz operator to the S = 1/2 single-transition operator, which shows that overtone NMR resonates at twice the Larmor frequency, i.e., 𝜔 𝐷𝑄 = 2(𝜔 0 + 𝜔 𝑄 (2) ).

In the rotating frame

To simplify calculations, we can define a frame R rotating at the irradiation frequency ωirr/2 around the z axis of the D frame. The rotation matrix can be described in the spin S = 1/2 doublequantum transition operator 𝑅 = exp(𝑖𝜔 𝑖𝑟𝑟 𝑡𝑆 𝑧 𝐷𝑄 ) for the transformation from the D to the R frame. In this frame, the internal spin Hamiltonian becomes

𝐻 𝑆 𝑅 = 𝛺 𝐷𝑄 𝑆 𝑧 𝐷𝑄 = (𝜔 𝐷𝑄 -𝜔 𝑖𝑟𝑟 )𝑆 𝑧 𝐷𝑄 (28)
where 𝛺 𝐷𝑄 is the resonance frequency offset of the DQ transition.

The detection operator

By combining Eqs. 15, 18 and 20, the detection operator in the D frame can be written as

𝑂 𝑑𝑒𝑡 𝐷 = (𝟏 -𝜀𝑉)𝑂 𝑑𝑒𝑡 𝐿 (𝟏 + 𝜀𝑉) = 𝑂 𝑑𝑒𝑡 𝐿 + 𝜀[𝑂 𝑑𝑒𝑡 𝐿 , 𝑉] -𝜀 2 𝑉𝑂 𝑑𝑒𝑡 𝐿 𝑉 (29)
Again for simplicity, we use the single-transition operators to describe the DQ transition of 14 N OT with the detection operator for the quadrature-detected 14 N OT signal given by

𝑂 𝑑𝑒𝑡 𝐷 = -𝜀(𝐴 2,1 𝑄,𝐿 sin𝜃 𝐶 + 𝐴 2,2 𝑄,𝐿 cos𝜃 𝐶 )𝑆 - 𝐷𝑄 = 𝜉 * 𝑆 - 𝐷𝑄 (32)
where the overtone parameter ξ is defined as

𝜉 = 𝜀(𝐴 2,-1 𝑄,𝐿 sin𝜃 𝐶 -𝐴 2,-2 𝑄,𝐿 cos𝜃 𝐶 ) (33)

The rf Hamiltonian

Let us now consider the case of overtone excitation with rf irradiation near twice the Larmor frequency. We assume the rf Hamiltonian to be small with respect to 𝐻 𝑆 𝐿 , and hence it transforms into the D frame as

𝐻 𝑟𝑓 𝐷 = 𝑇 -1 𝐻 𝑟𝑓 𝐿 𝑇 (34)
We can follow the same procedure used for the previous derivation of Eq. 31 and keep only the DQ elements for overtone excitation:

𝐻 𝑟𝑓 𝐷 = 2𝜔 1 cos(𝜔 𝑖𝑟𝑟 𝑡 + 𝜙) (𝜉𝑇 2,2 𝑄 + 𝜉 * 𝑇 2,-2 𝑄 ) (35)
The rapid oscillation of the rf Hamiltonian in the laboratory frame can be reduced by using the rotating frame R previously defined in Section 2.2.3:

𝐻 𝑟𝑓 𝑅 = 𝑅 -1 𝐻 𝑟𝑓 𝐷 𝑅 = 𝜔 1 [(1 + exp[-𝑖(2𝜔 𝑖𝑟𝑟 𝑡 + 𝜙)])𝜉𝑇 2,2 𝑄 +(1 + exp[𝑖(2𝜔 𝑖𝑟𝑟 𝑡 + 𝜙)])𝜉 * 𝑇 2,-2 𝑄 ] (36)
Indeed, when we neglect the effects of the non-resonant rotating component of the rf field, this

Hamiltonian becomes time-independent and can be written as

𝐻 𝑟𝑓 𝑅 ≈ 𝜔 1 (𝜉𝑒 -𝑖𝜙 𝑆 + 𝐷𝑄 + 𝜉 * 𝑒 𝑖𝜙 𝑆 - 𝑫𝑸 ) = 2𝜔 1 |𝜉|exp(-𝑖𝜙 𝐷𝑄 𝑆 𝑧 𝐷𝑄 )𝑆 𝑥 𝐷𝑄 exp(𝑖𝜙 𝐷𝑄 𝑆 𝑧 𝐷𝑄 ) (37)
where

𝜙 𝐷𝑄 = 𝜙 + arg(𝜉 * ) (38)
Eqs. 28, 32 and 37 summarize the results of the 14 N OT theory under static conditions. 14 N OT NMR can be treated as conventional NMR of a fictitious spin S = 1/2 nucleus with a single DQ overtone transition between two energy levels. Both the rf excitation and detection is scaled by the overtone parameter in Eq. 33.

In the R frame, the equilibrium state is defined by the density operator 𝜎 𝑅 (0) ∝ 2𝑆 𝑧 𝐷𝑄 when considering only the zeroth-order term in Eq. 15. The time evolution after pulse excitation is given by

𝜎 𝑅 (𝑡 > 𝜏 𝑝 ) = exp[-𝑖𝐻 𝑆 𝑅 (𝑡 -𝜏 𝑝 )] exp(-𝑖𝐻 𝑟𝑓 𝑅 𝜏 𝑝 ) 𝜎 𝑅 (0) × exp(𝑖𝐻 𝑟𝑓 𝑅 𝜏 𝑝 ) exp[𝑖𝐻 𝑆 𝑅 (𝑡 -𝜏 𝑝 )] ( 39
)
where τp is the pulse length and t denotes the time elapsed from the start of the pulse. 𝑆 𝑥 𝐷𝑄 , 𝑆 𝑦 𝐷𝑄 and 𝑆 𝑧 𝐷𝑄 are three cyclically commutating spin S = 1/2 operators for the double-quantum transition.

Using Eqs. 28 and 37, we obtain

𝜎 𝑅 (𝑡 > 𝜏 𝑝 ) = 2cos(2|𝜉|𝜔 1 𝜏 𝑝 )𝑆 𝑧 𝐷𝑄 -2sin(2|𝜉|𝜔 1 𝜏 𝑝 ) × (sin[𝛺 𝐷𝑄 (𝑡 -𝜏 𝑝 ) -𝜙 𝐷𝑄 ] 𝑆 𝑥 𝐷𝑄 -cos[𝛺 𝐷𝑄 (𝑡 -𝜏 𝑝 ) -𝜙 𝐷𝑄 ] 𝑆 𝑦 𝐷𝑄 ) (40)
Assuming that off-resonance effects can be neglected, i.e., the 𝐻 𝑟𝑓 𝑅 Hamiltonian is large with respect to 𝐻 𝑆 𝑅 , the 14 N OT signal can be calculated in the R frame as:

𝑠(𝑡 > 𝜏 𝑝 ) ∝ Tr[𝑂 𝑑𝑒𝑡 𝑅, † 𝜎 𝑅 (𝑡 > 𝜏 𝑝 )] = |𝜉|sin(2|𝜉|𝜔 1 𝜏 𝑝 )exp(𝑖[𝜔 𝐷𝑄 (𝑡 -𝜏 𝑝 ) + 𝜔 𝑖𝑟𝑟 𝜏 𝑝 -𝜙 -𝜋/2]) (41)
The overtone signal detection (Eq. 34) and rf excitation (Eq. 37) are scaled down by the same overtone parameter, |ξ|, which is proportional to the ratio between the quadrupolar coupling constant and the Larmor frequency: 𝜀 = 𝜔 𝑄 𝜔 0 ⁄ . The overtone parameter is anisotropic through the single-and double-quantum elements of the quadrupole Hamiltonian, 𝐴 2,-1 𝑄,𝐿 and 𝐴 2,-2 𝑄,𝐿 , respectively (Eq. 33), and therefore it depends on the molecular/crystallite orientation (Eq. 5). ξ is also sensitive to the rf coil orientation with respect to the magnetic field B0 as defined by the angle θC in Eq. 33. These features are unique to 14 N OT NMR in which the overtone parameter plays a central role.

Theory for rotating samples

Periodically time-dependent spin Hamiltonian

We now consider a solid rotating in the L frame. The internal spin Hamiltonian, 𝐻 𝑆 𝐿 , becomes time modulated, including the off-diagonal elements of the quadrupolar interaction that make the overtone transition detectable,

𝐻 𝑄 𝐿 (𝑡) = 𝜔 𝑄 [ 𝐴 2,0 𝑄,𝐿 (𝑡)/√6 𝐴 2,-1 𝑄,𝐿 (𝑡) √2 ⁄ 𝐴 2,-2 𝑄,𝐿 (𝑡) -𝐴 2,1 𝑄,𝐿 (𝑡) √2 ⁄ -𝐴 2,0 𝑄,𝐿 (𝑡)√2/3 -𝐴 2,-1 𝑄,𝐿 (𝑡) √2 ⁄ 𝐴 2,2 𝑄,𝐿 (𝑡) 𝐴 2,1 𝑄,𝐿 (𝑡) √2 ⁄ 𝐴 2,0 𝑄,𝐿 (𝑡)/√6 ] (42)
The spatial tensor components, 𝐴 2,𝑚 𝑄,𝐿 (t), can be obtained from those, 𝐴 2,𝑚 ′ 𝑄,𝑃 , in the P frame of the EFG tensor by two consecutive rotations, one from the laboratory to the rotor frame with the Euler angles (-ωrt, θC, 0) and the second from the rotor to the PAS frame (αPR, βPR, γPR):

𝐴 2,𝑚 𝑄,𝐿 (𝑡) = ∑ 𝐴 2,𝑚 ′ 𝑄,𝑃 𝐷 𝑚 ′ 𝑘 2 (𝛼 𝑃𝑅 , 𝛽 𝑃𝑅 , 𝛾 𝑃𝑅 )𝐷 𝑘𝑚 2 (-𝜔 𝑟 𝑡, 𝜃 𝐶 , 0) 2 𝑘,𝑚 ′ =-2 (43)
where

𝐷 𝑚 ′ 𝑘 2 (𝛼 𝑃𝑅 , 𝛽 𝑃𝑅 , 𝛾 𝑃𝑅 ) = exp(-𝑖𝛼 𝑃𝑅 ) 𝑑 𝑚 ′ 𝑘 2 (𝛽 𝑃𝑅 ) exp(-𝑖𝑘𝛾 𝑃𝑅 ) (44)
𝐷 𝑘𝑚 2 (-𝜔 𝑟 𝑡, 𝜃 𝐶 , 0) = exp(𝑖𝑘𝜔 𝑟 𝑡) 𝑑 𝑘𝑚 2 (𝜃 𝐶 ) (45)
We assume here that the spinning axis and rf coil axis coincide at angle θC with respect to the B0 field, as it is usually the case in MAS probes. For rotating solids, The 𝐴 2,𝑚 𝑄,𝐿 components can thus be expanded as a Fourier series with respect to the angular rotation frequency ωr,

𝐴 2,𝑚 𝑄,𝐿 (𝑡) = ∑ 𝑎 𝑚 𝑘 exp(𝑖𝑘𝜔 𝑟 𝑡) 2 𝑘=-2 (46)
with

𝑎 𝑚 𝑘 = ∑ 𝐴 2,𝑚 ′ 𝑄,𝑃 𝐷 𝑚 ′ 𝑘 2 (𝛼 𝑃𝑅 , 𝛽 𝑃𝑅 , 𝛾 𝑃𝑅 )𝑑 𝑘𝑚 2 (𝜃 𝐶 ) 2 𝑚 ′ =-2 . ( 47
)
The quadrupolar Hamiltonian can thus be written as a sum of five matrices ℎ 𝑄 𝑘 ,

𝐻 𝑄 𝐿 (𝑡) = 𝜔 𝑄 ∑ ℎ 𝑄 𝑘 exp(𝑖𝑘𝜔 𝑟 𝑡) 2 𝑘=-2 (48)
with

ℎ 𝑄 𝑘 = [ 𝑎 0 𝑘 /√6 𝑎 -1 𝑘 √2 ⁄ 𝑎 -2 𝑘 -𝑎 1 𝑘 √2 ⁄ -𝑎 0 𝑘 √2/3 -𝑎 -1 𝑘 √2 ⁄ 𝑎 2 𝑘 𝑎 1 𝑘 √2 ⁄ 𝑎 0 𝑘 /√6 ] (49)
Here we assume that ωr < ωQ, and hence we must consider the terms with k ≠ 0 in Eq. 48. Similar to the static case, we can focus only on the off-diagonal quadrupolar elements of Eq. 49 that make 14 N OT NMR possible.

Floquet's theory and the diagonal tilted transformation

Floquet's theory is invoked to treat the periodic, modulated perturbations.

Assuming 𝜀 = 𝜔 𝑄 𝜔 0 ⁄ < 1 and 𝜔 𝑟 ≪ 𝜔 𝑄 , the T(t) operator that diagonalizes the Hamiltonian can be obtained using a first-order perturbation treatment of the Floquet Hamiltonian [START_REF] Shirley | [END_REF]74 .

𝑇(𝑡) = 𝟏 + 𝜀 ∑ 𝑉 𝑘 exp(𝑖𝑘𝜔 𝑟 𝑡) 𝑘=±1,±2 (51)
where

𝑉 𝑘 = [ 0 𝑎 -1 𝑘 √2 ⁄ 𝑎 -2 𝑘 2 ⁄ -𝑎 1 𝑘 √2 ⁄ 0 -𝑎 -1 𝑘 √2 ⁄ 𝑎 2 𝑘 2 ⁄ 𝑎 1 𝑘 √2 ⁄ 0 ] (52)

The internal spin Hamiltonian

The 𝐻 𝑆 𝐷 in Eq. 50 of the Floquet theory is related to the average Hamiltonian 〈𝐻〉

〈𝐻〉 = 𝑇(0)𝐻 𝑆 𝐷 𝑇(0) -1 (53)
Thus it can be calculated as in the static case (Eqs. 21 to 27), retaining only the diagonal and timeindependent terms

𝐻 𝑆 𝐷 = 2(𝜔 0 + 𝜔 ̅ 𝑄 (2) )𝑆 𝑧 𝐷𝑄 = 𝜔 ̅ 𝐷𝑄 𝑆 𝑧 𝐷𝑄 (54)
Here, 𝜔 ̅ 𝐷𝑄 is the time-averaged resonance frequency of the DQ transition. The time-averaged second-order quadrupole frequency (see Eq. 26) is given by

𝜔 ̅ 𝑄 (2) = 𝜔 𝑄 2 𝜔 0 ∑ (-𝑎 1 -𝑘 𝑎 -1 𝑘 + 𝑎 2 -𝑘 𝑎 -2 𝑘 ) 2 𝑘=-2 (55)
This frequency can also be expressed as a function of second-and fourth-rank spatial tensors 74 .

The diagonal tilted transformation can also induce a geometric phase term or so-called Berry's phase from the time dependent Hamiltonian to the averaged peak position for rotating samples. Berry's phase can be visualized as the solid angle encompassed by the sweeping quantization axis of the cyclic Hamiltonian. 75 The accumulation of Berry's phase can result in a shift proportional to the spinning speed and sensitive to the spinning direction with respect to the quantization axis as it was experimentally demonstrated by Nuclear Quadrupole Resonance (NQR) of a rotating single crystal at and near zero-field. 76 In the case of 14 N overtone at high fields, the quatization axis is just slightly off from the main magnetic field. The encompassed solid angle or the Berry's is small, in the order of (𝜔 𝑄 𝜔 0 ⁄ ) 2 4𝜋 ⁄ . Therefore the resulting shift can be estimated in the order of (𝜔 𝑄 𝜔 0 ⁄ ) 2 𝜔 𝑟 4𝜋 ⁄ , 𝜔 𝑟 𝜔 𝑄 ⁄ smaller the second-order quadrupolar shift in Eq. ( 55).

The Berry's phase may become observable at high fields under fast spinning.

The detection operator

The detection operator in the D frame, which is transformed into the L frame by the matrix T(t), is given by

𝑂 𝑑𝑒𝑡 𝐷 = 𝑇(𝑡) -1 𝑂 𝑑𝑒𝑡 𝐿 𝑇(𝑡) (56)
Following the same procedure we have used to derive Eq. 32 in the static case, we obtain for the overtone detection operator under sample rotation,

𝑂 𝑑𝑒𝑡 𝐷 = 𝜉 𝑟𝑜𝑡 * (𝑡) 𝑆 - 𝐷𝑄 (57)
where

𝜉
which provides the following expressions

𝜒 0 (𝜃 𝐶 ) = - 3 2 √ 3 2 cos 𝜃 𝐶 sin 2 𝜃 𝐶 𝜒 1 (𝜃 𝐶 ) = -𝜒 -1 (𝜋 -𝜃 𝐶 ) = 2cos 2 (𝜃 𝐶 /2)(1 + 3cos𝜃 𝐶 )sin 3 (𝜃 𝐶 /2) (62) 𝜒 2 (𝜃 𝐶 ) = -𝜒 -2 (𝜋 -𝜃 𝐶 ) = -(2 + 3cos𝜃 𝐶 )sin 4 (𝜃 𝐶 /2)
Equations ( 58), ( 60) and (61) show that for rotating samples the overtone parameter have five modulating components. The relative amplitudes among the five components are constants for all orientation in powder samples which are determined only by the angle of spinning and coil axis with respect to the magnetic field. In the next section, the numerical values of the constant show that either 𝑘 = 2 or -2 component is dominant under magic-angle spinning leading to apparent overtone peak shifting at twice of the spinning frequency. This is the key finding from theory which explains the intriguing feature of 14 N overtone NMR observed experimentally under MAS. 63 Previous theoretic studies 57,61,62 have not gone far enough to reach this conclusion.

The rf Hamiltonian

The rf spin Hamiltonian can also be expressed in the D frame as

𝐻 𝑟𝑓 𝐷 = 𝑇(𝑡) -1 𝐻 𝑟𝑓 𝐿 𝑇(𝑡) (63)
Using the same procedure as in the static case, we can express the rf Hamiltonian in a frame R rotating at the irradiation frequency ωirr/2 around the z-axis of the D frame (section 2.2.3).

Neglecting the non-resonant rotating component of the rf field, the rf spin Hamiltonian in such a R frame can be written into a form similar to Eq. 37 derived in the static case as

𝐻 𝑟𝑓 𝑅 ≈ 𝜔 1 (𝜉 𝑟𝑜𝑡 (𝑡, 𝜃 𝐶 )𝑒 -𝑖𝜙 𝑆 + 𝐷𝑄 + 𝜉 𝑟𝑜𝑡 * (𝑡, 𝜃 𝐶 )𝑒 𝑖𝜙 𝑆 - 𝐷𝑄 ) (64)
The sample spinning adds additional modulations to the oscillating rf field through the overtone parameter. With short rf pulses in the linear excitation regime, the contributions from the five modulating components are additive. We can express the overtone signal as

𝑠(𝑡 > 𝜏 𝑝 ) ∝ ∑ |𝜉 𝑘 (𝜃 𝐶 )|sin(2|𝜉 𝑘 (𝜃 𝐶 )|𝜔 1 𝜏 𝑝 ) 2 𝑘=-2 × exp(𝑖[(𝜔 ̅ 𝐷𝑄 + 𝑘𝜔 𝑟 )(𝑡 -𝜏 𝑝 ) + (𝜔 𝑖𝑟𝑟 + 𝑘𝜔 𝑟 )𝜏 𝑝 -𝜙 -𝜋/2]) (65)
The overtone signal of rotating solids hence contains five modulating components, which lead to five resonances separated by the spinning frequency. Their appearance is similar to the spinning sidebands often seen in solid state NMR of rotating samples, however with a fundamental difference. Conventional spinning sideband intensities depend on the ratio between the magnitude of the observed NMR frequency modulation and the spinning frequency, and hence they diminish with faster spinning. On the contrary, the five overtone peaks do not result from the modulation of the 14 N second-order quadrupole interaction, which affects the resonance frequency of the overtone signal (Eq. 54), but from the modulation of the total quadrupole interaction, which affects the efficiency of excitation and detection of the overtone transition under sample rotation. The intensities of the five components are hence independent of the spinning frequency. Moreover, Eq. 62 shows that the relative intensity of the five overtone sidebands depends on the angle of the rotor axis, C, which will be discussed later. When the CSA or the 14 N second-order quadrupole interaction is larger than ωr, the modulation of the overtone DQ frequency leads to extra spinning sidebands in addition to the five overtone components. These sidebands behave like conventional ones and thus disappear with higher spinning frequencies like those typically used experimentally in 14 N OT NMR.

Results and discussions

The 14 N OT NMR theory presented above centers on the overtone parameter, ξ or ξk for static or rotating conditions, respectively, which depends on the quadrupole interaction, the crystallite orientation, and the spinning/rf coil axis angle, C, with respect to B0.

First, ξ and ξk scale both rf excitation and signal detection, and are proportional to the ratio between the quadrupole and Zeeman interactions through the parameter 𝜀 = 𝜔 𝑄 𝜔 0 ⁄ (Eqs. 33 and 59). Because of the scaling factor, 14 N OT NMR may look a priori less favorable at high magnetic fields. However, there are several advantages with increasing B0 magnetic field: (i) increase in spin polarization due to the Boltzmann factor, (ii) decrease of second-order quadrupolar line broadening, (iii) increase sensitivity due to the inductively detected NMR signal being proportional to frequency, (iv) increase in separation between various resonances through chemical shift differences. The overall overtone sensivity still increases with the magnetic field considering all these factors. 77 Second, ξ and ξk depend on the molecular orientation through the 𝐴 2,-1 𝑄,𝐿 and 𝐴 2,-2 𝑄,𝐿 (see Eq.

33)

or 𝑎 -1 𝑘 and 𝑎 -2 𝑘 (see Eq. 59) terms in static or rotating samples, respectively; which in turn depend on the angle between the coil and the magnetic field, θC. The consequences will be discussed further below for the cases of static and rotating samples.

Static samples

Fig. 2a shows the distribution in amplitude and phase of the overtone parameter ξ in a static powder sample (Eq. 33). We have used the 14 N parameters of glycine and a field of B0 = 11.74 T for comparison with the previously reported simulations and experimental results. 63,64 In this case, |ξ| ranges from 0 to 1.5% and therefore both the overtone rf excitation and detection are much reduced. Furthermore, Fig. 2a also shows that the phase of this complex parameter is randomly distributed as a result of the distributed orientations of the quadrupolar tensor with respect to the magnetic field. This random phase distribution may lead to a confusing physical picture as opposed to conventional NMR. In the latter case: (i) rf excitation and signal detection are usually coherent for all spins; (ii) the polarizations from all crystallites align with the magnetic field before excitation and they remain coherent after excitation; and (iii) this coherent alignment leads to an overall NMR signal with almost no cancellation. As a result, we often relate the spin polarization directly with the NMR signal. All these facts are no longer true for 14 N OT NMR due to the random phase distribution of ξ. One may then wonder why it is still possible to detect 14 N OT signals. Indeed, if the overtone DQ polarization was excited uniformly, in a similar way as the excitation of multiple-quantum coherences in MQMAS, then a powder average of  during overtone detection would annihilate the signal. The key point to overtone NMR is that the same overtone parameter  and its phase distribution apply for signal detection and rf excitation at twice the Larmor frequency. Therefore, the two phase distributions cancel each other and result in a 14 N OT signal proportional to the magnitude of ξ (see Eq. 41), which is observable even for a powder sample.

Nevertheless, the excitation and detection profiles depend on |ξ|, which makes the line shape sensitive to the excitation pulse length. In the short pulse limit, sin(2|ξ|1p)  2|ξ|1p, and the overtone peak intensity is proportional to the square of the overtone parameter:

s(ω)  |ξ| 2 (66)
Figs. 2b and2c show that the short-pulse static line shape varies with the angle C between the rf coil and B0, a feature unique to 14 N OT NMR. Overall, only the magnitude of the overtone parameter  needs be considered. For 14 N OT excitation, the spin dynamics is then almost identical to that of spin S = 1/2 nuclei, except for the rf scaling and its anisotropic angular dependence. However, as a general rule, the effective overtone rf field is proportional to 1|ξ| (Eq. 37), and hence to ω1ωQ/ω0.

The practical consequences of the scaled down effective rf field lead to (i) long pulse durations with small excitation bandwidths, (ii) difficulties in generating 14 N OT spin-echoes without loss of efficiency and line shape distortions, and (iii) non-uniform excitation due to the anisotropic angular dependence.

Magic-Angle Spinning samples

For rotating solids, MAS probes are usually used for 14 N OT NMR to cancel the CSA and dipolar interactions, and both the spinning and coil axes are co-linear and at the magic-angle with respect to the magnetic field B0, i.e., C = M = 54.74°. From Eq. 62, we obtain the following numerical values for the amplitude of the five overtone components under MAS, χk (θM) = (0.11, 0.27,0.23, -0.25, -0.88) (k = 2, 1, 0, -1, -2) (67)

In the short pulse limit, the overtone peak intensities of the five resonances are proportional to the square of the overtone parameter:

ssbk  k 2 . ( 68
)
Thus, the relative amplitudes of the five 14 N OT 'spinning sidebands' (ssbs) are given by ssbk (M) = (0.02, 0.09, 0.07, 0.09, 1.00) (k = 2, 1, 0, -1, -2) (

The term 'spinning sidebands' is retained as these peaks also shift with the spinning frequency.

However, there are fundamental differences in the physical origin and behavior of the 14 N OT ssbs as compared to spinning sidebands observed in conventional MAS experiments. For conventional NMR, the center-band is usually the dominant peak under fast spinning, whereas for overtone NMR it is the k = -2 sideband. The other overtone ssbs are at least an order of magnitude smaller, which makes the most prominent overtone signal appear to shift by twice the spinning frequency.

Furthermore, the relative intensity of the ssbs in overtone spectra are mostly independent of spinning frequency.

It should be noted that if the spinning axis is inverted with respect to B0, i.e., M → 180° -M or equivalently r → -r, the relative amplitudes of the overtone ssbs, which are proportional to k 2 in the short pulse regime (Eq.66), reverse in order (Eq. 62), ssbk ( -M) = (1.00, 0.09, 0.07, 0.09, 0.02) (k = 2, 1, 0, -1, -

The k = 2 sideband becomes dominant and the main signal shifts by twice the spinning frequency in the opposite direction. 14 N OT NMR is a rare case where the spectra are sensitive to the sense of spinning with respect to the magnetic field.

First, the most prominent overtone peak shifts at twice the spinning frequency, while its energy level remains at the center band position, i.e., at the time averaged frequency over one rotor period.

By placing the rf carrier frequency onto the main overtone peak position, i.e., at the -2ωr sideband, is the rf irradiation on-or off-resonance? The answer is that the effective excitation is on-resonance.

Although the rf irradiation is offset by 2ωr, the modulation of the rf field by the 2ωr component in Eq. 71 effectively makes the overtone irradiation frequency match the overtone DQ transition.

Second, the five modulating k components of Eq. 71 can all excite the overtone transition despite their frequency offsets. How much does each of the five modulating k components mutually contribute to other overtone ssbs? In other words, can the dominant modulating component be used to excite the other smaller overtone sidebands? The answer to the second question is that only excitation from the same modulating component contributes to the sideband being observed. The reason lies in the effect of the rotor angle γPR on the overtone rf field, which appears in Eq. 71 as the phase kγPR. In the linear excitation regime, the excitations from other components have a nonvanishing γPR angular dependence given by exp(𝑖[𝑘 𝑒𝑥𝑐 -𝑘 𝑑𝑒𝑡 ]𝛾 𝑃𝑅 ), which is annihilated by powder averaging; kexc and kdet are the modulating sideband orders for the excitation and detection, respectively. Thus, only the excitation from the same sideband component (i.e., kexc = kdet) contributes to the overtone peak being observed. In the short pulse limit, the peak intensities of the overtone sidebands are given by the square of the relative amplitude k which effectively amplifies the differences in relative intensity and makes the ±2ωr peak appearing more dominant under MAS.

Fig. 4 shows experimental 14 N OT MAS spectra of glycine at r = 10 kHz, with the rf carrier frequency set on each of the five observed sidebands. The 14 N OT peak intensities agree with the simulated results for the relative sideband intensities in Fig. 3. In order to confirm the dependence of the spinning induced 14 N OT MAS shift on the sense of spinning relative to the magnetic field,

we have searched all possible magnet and probe combinations available to us and found that all vertical-bore superconducting NMR magnets we have access to have their magnetic fields pointing upwards. Most of the MAS probes are equipped with spinning modules from Bruker and Revolution NMR which (coincidentally?) spin samples in the same direction (counterclockwise when looking down the coil/stator axis). Only one of our decommissioned Doty probes spins the samples in the opposite direction which allowed us to confirm experimentally that the spinning induced shift of the main overtone peak is reversed in the 'opposite' spinning direction. Agreement has also been observed from 14 N OT MAS NMR spectra acquired later using JEOL probes which also spin in the clockwise direction (not shown). Experimentally, long rf pulses are usually used in order to observe sufficient signal intensity. The narrow excitation bandwidth of long pulses makes the simultaneous observation of all five sidebands difficult. The five sidebands in Fig. 4 were acquired individually by placing the overtone rf frequency on each of the marked peak positions in Fig. 4. The National High Magnetic Field Laboratory has recently commissioned a 36 Tesla series-connected-hybrid magnet with field homogeneity and stability suitable for high-resolution solid-state NMR experiments. 10 The powered magnet can be ramped to the full field strength in either direction in approximately 30 minutes. Thus, 14 N OT MAS NMR spectra can be acquired back-to-back spinning in the same direction but with opposite field orientations in about an hour.

Fig. 5 shows the 14 N OT MAS NMR spectra of glycine with the magnetic field in opposite directions.

In addition, 14 N OT MAS spectra were acquired at two spinning frequencies to show the spinning induced shift of the 2ωr overtone sideband. The lower peak intensities at ωR/2π = 5 kHz are due to the fact that the single-channel probe used relies solely on MAS to average the dipolar coupling to the protons. Low spinning causes less efficient decoupling, contributing to broader overtone lines and lower intensities. The results confirm again the opposite spinning induced shift of 14 N overtone peaks with respect to the magnetic field direction.

Fig. 5. Experimental 14 N OT MAS NMR spectra of glycine recorded at 35.2 T using the series-connected-hybrid (SCH) magnet at the National High Magnetic Field Laboratory. A 3.2 mm home-built MAS probe with 36 l sample volume was used for the measurement. A 400 s pulse with approximately 100 kHz rf field was used for the overtone excitation. The frequency offset was placed at the main overtone peaks. 8192 scans with 0.1 s recycle delay were acquired for each spectrum. Two sets of spectra were acquired back-to-back by reversing the direction of the magnetic field for ωR/(2π) = 5 and 10 kHz spinning frequencies.

Conclusions

We have shown than overtone NMR of 14 N nuclei, or more generally of any spin S = 1 nucleus, can be described by an overtone parameter in the case of static or rotating samples. This parameter, which scales down both the effective rf field and the detected signal, depends on the quadrupolar coupling interaction and the crystallite orientation, and is inversely proportional to the magnetic field. The spin dynamics of overtone excitation can be treated as a fictitious spin S = 1/2 nucleus represented by a two-level overtone transition and a rf field scaled down by the overtone parameter. For rotating samples, the overtone parameter is modulated by the spinning, which gives rise to five components or spinning sidebands. The relative amplitudes of the five sidebands are only determined by the orientation of the rf coil and the spinning axis with respect to the magnetic field. For a solenoid coil and spinning axis along the magic angle, the relative signal amplitudes are (0.02, 0.09, 0.07, 0.09, 1.00). Thus, the 2ωr sideband dominates and the main observable overtone peak appears shifted, at twice the spinning frequency. Reversing the spinning (or magnetic field) direction makes the peak shift in the opposite direction. The presented theory gives a physical explanation for the intriguing features observed in the overtone NMR spectra of rotating samples, and a simple formalism for the spin dynamics therein. The understanding and presented formalism will help in exploiting 14 N OT NMR for the highly abundant nitrogen isotope.

Fig. 2 .

 2 Fig. 2. Simulations of 14 N OT NMR data under static conditions with B0 = 11.74 T (ω0,14N/(2π) = 36.118 MHz), a short pulse, and the parameters of glycine (CQ = 1.18 MHz, ηQ = 0.53, δiso = 6 ppm). (a) Distribution of the magnitude and

Fig. 3

 3 Fig.3shows simulations of the 14 N OT NMR line shape and relative sideband intensities in

Fig. 3 . 2 ×

 32 Fig. 3. Simulations of 14 N OT short-pulse spectra under rotation at various θC angles of the spinning/coil axis, with B0 = 11.74 T, ω0,14N/(2π) = 36.118 MHz, ωR/(2π) = 8 kHz, and the parameters of glycine (see Fig. 2 caption). The overtone peaks were scaled by |ξk| 2 .

Fig. 4 .

 4 Fig. 4. Experimental 14 N OT MAS spectra of glycine recorded at 19.6 T, with ωR/(2π) = 10 and 3.5 kHz using a 4 mm Bruker and 5 mm Doty MAS probes with approximately 70 and 110 l sample volume, respectively.The rf amplitude ω1/(2π) was calibrated using a D2O sample and was approximately 60 and 20 kHz for the Bruker and Doty probe, respectively. The deuterium Larmor frequency is within 6% of the 14 N overtone frequency. 100 and 50 s long excitation pulses were used for the Bruker and Doty probes with frequency offsets set on resonance with respect to the overtone peaks indicated by the arrows. The recycle delay was 0.5 s and the number of scans were 1024 for measuring the main -2 sideband, 32768 for the other sidebands of the Bruker 4 mm experiment and 102400 for the Doty 5 mm experiment. The two probes spin the samples in opposite directions/senses with respect to B0. The spectra recorded with the Bruker probe were acquired using five different carrier frequencies denoted by the arrows. Only one spectrum with the frequency set at the expected main overtone peak position was acquired using the Doty probe due to the weak signal caused by the low rf field. The results confirm the second spinning sideband as the main overtone

A. Brinkmann and L. A. O'Dell, Solid State Nucl. Magn. Reson. 84, 34 (2017).

Acknowledgements

This work was supported by the National High Magnetic Field Laboratory through National Science Foundation Cooperative Agreement (DMR-1644779) and by the State of Florida. Development of the SCH magnet and NMR instrumentation was supported by NSF (DMR-1039938 and DMR-0603042). JT, OL and JPA also acknowledge contract CEFIPRA n°85208-E, PRC CNRS-NSFC and ANR-17-ERC2-0022 (EOS). HN acknowledges the Univ. Lille and Région Hauts-de-France for his PhD grant. BH acknowledges National Natural Science Foundation of China for Excellent Young Scholars (Grant No. 21522303). YN acknowledges JSPS KAKENHI (Grant number JP17H03043).