Maud Van Den Broeke

Nicholas Paparoidamis

Engaging in or Escaping Co-creation? An Analytical Model

Keywords: customer co-creation, flexibility, product development effort

Customer co-creation, the practice of involving the customer in a firm's new product development, has received increasing attention. We develop a unique analytical model to study co-creation and examine the conditions under which co-creation is economically beneficial for both the customer and the firm. In our model, the customer and the firm determine (simultaneously or sequentially) their innovation share in the co-created product, directly affecting their share in the development costs and the final co-created product quality. The firm decides the product price (affecting customer demand, firm revenues and customer purchasing costs) and its level of manufacturing flexibility (affecting its unit production cost and fixed investment). Our model generates new and important insights. We show that when the consumer surplus and firm profit are both positive, both the customer and firm want to "engage" in co-creation, and when either the customer or the firm obtains a negative economic value from co-creation they prefer to avoid co-creation. We find that co-creation thrives when customers are more sensitive to quality and less sensitive to price, the environment is less investment intensive, and demand for the product is high.

Introduction

Customer co-creation, also called "co-design" or "co-development", is a product development approach in which customers actively take part in the design of their product [START_REF] Prahalad | Co-creation experiences: The next practice in value creation[END_REF][START_REF] Wallace | Putting customers in charge of design[END_REF]. Firms have turned to co-creation to better identify and fulfill customer needs, increasing new product success [START_REF] Franke | Testing the value of customization: when do customers really prefer products tailored to their preferences[END_REF][START_REF] Randall | Research note: User design of customized products[END_REF][START_REF] Zhang | Examining the mechanism of the value co-creation with customers[END_REF]. Several success stories of business-to-consumer (B2C) co-creation exist, such as MiAdidas and NIKEiD, in which customers can design their personalized shoes online using a variety of materials and colors; Threadless, which markets customer-designed T-shirts; and Lego Ideas, in which customers can suggest new Lego sets online [START_REF] Piller | Mass customization at adidas: Three strategic capabilities to implement mass customization[END_REF][START_REF] Schreier | The innovation effect of user design: Exploring consumers' innovation perceptions of firms selling products designed by users[END_REF].

A wealth of research has discussed the advantages of co-creation, assuming that cocreation creates value for both the customer and the producer [START_REF] Leclercq | Ten years of value cocreation: An integrative review[END_REF][START_REF] Mahr | The value of customer cocreated knowledge during the innovation process[END_REF][START_REF] Payne | Managing the co-creation of value[END_REF]. However, little research attention has been devoted to the case when co-creation becomes unfavorable for either the customer or the firm [START_REF] Gemser | Co-creation with customers: An evolving innovation research field[END_REF][START_REF] Weber | Does customer co-creation really pay off? an investigation into the firm's benefits from customer involvement in new product and service development[END_REF]. In some companies, co-creation does not deliver the desired outcome (e.g., the failure of Mattel's My Design Barbie program). Therefore, in this article, we answer the following main question: Under which conditions is co-creation beneficial for both the firm and customer? Specifically, we examine when co-creation is no longer economically beneficial for either the customer or the firm or both? (i.e., leads to a negative consumer surplus or a negative firm profit), in which environments co-creation thrives (i.e., the likelihood of mutually beneficial co-creation and the co-created product quality increase)? and how innovation efforts should be shared between the customer and the firm?

To answer these questions, we develop a unique analytical model. We model a firm collaborating with its customer for the development of a new product (i.e., a B2C setting). We consider the consumer surplus and firm profit -two well-accepted economic measurements, embedded in welfare economics [START_REF] Hausman | Exact consumer's surplus and deadweight loss[END_REF][START_REF] Willig | Consumer's surplus without apology[END_REF]) -to measure the economic benefit or gain after collaboration of the customer and firm respectively (these measurements are also used in, for instance, the work of [START_REF] Syam | Co-creation with production externalities[END_REF], [START_REF] Syam | Customized products: A competitive analysis[END_REF] and [START_REF] Baldwin | How user innovations become commercial products: A theoretical investigation and case study[END_REF]). We calculate the consumer surplus as the difference between the customer's willingness to pay and the price, for different quantities of a product, based on the (aggregate) demand curve, minus the development costs, while we calculate the firm's profit, seen as the producer surplus, as their revenues minus the production, fixed investment, and development costs.

On the one hand, the customer and firm decide (either simultaneously or sequentially) their innovation or (R&D) development share in the co-created product, affecting the final product quality and how development costs are shared. For example, at Lego Ideas, a wellknown co-creation initiative, the customer builds a prototype for a new Lego set, while the firm takes care of the later stages of design if the set is accepted for commercialization (Lego, 2018). At Adidas, customers are only involved in the later stage of the innovation cycle in which they can choose from a predefined range of options to customize their shoe, referred to as mass customization [START_REF] Alptekinoglu | Mass customization vs. mass production: Variety and price competition[END_REF][START_REF] Fogliatto | The mass customization decade: An updated review of the literature[END_REF][START_REF] Piller | Observations on the present and future of mass customization[END_REF].

Note that not only the sequence of who contributes first to the product development changes over different co-creation initiatives, but also the intensity of the customer's innovation contribution can range from being a source of information to being a real co-developer or creator (i.e., the customer innovation share can be small or large) [START_REF] Cui | Utilizing customer knowledge in innovation: antecedents and impact of customer involvement on new product performance[END_REF][START_REF] Hoyer | Consumer cocreation in new product development[END_REF].

On the other hand, the firm individually decides its level of manufacturing flexibility and the price charged for the product. The former affects the firm's variable production cost and fixed investment, while the latter affects the firm's revenues and the customer's purchasing cost. As an example of a manufacturing flexibility decision, Adidas has invested significantly in its Speedfactories to obtain a flexible automated digital manufacturing process to efficiently produce customized shoes [START_REF] Kim | Adidas' vision for the future: Personalization, fast[END_REF]. Note that in our model we consider an endogenous demand and a monopolistic firm.

Our analytical B2C co-creation model is an important contribution to the literature. It answers the call of [START_REF] Syam | Co-creation with production externalities[END_REF] for more economic models on co-creation, as conventionally co-creation has been studied using experimental methods, case studies, surveys, or conceptual frameworks, and has been looked at from a business-to-business perspective [START_REF] Gemser | Co-creation with customers: An evolving innovation research field[END_REF][START_REF] Leclercq | Ten years of value cocreation: An integrative review[END_REF]. Our model is also the first to include firm flexibility. It allows to study co-creation from a general perspective, and it forms a reference base for future co-creation studies.

Our results contribute to the literature in three ways. First, we examine the conditions under which co-creation is beneficial for both the co-creating customer and the firm. We partition the firm's manufacturing flexibility-joint co-created product quality space in different regions where co-creation is mutually beneficial, where both the customer and the firm want to escape co-creation (i.e., both parties obtain a negative economic value from co-creating), and where either the customer or the firm wants to escape co-creation (one of the parties obtains a negative value from co-creating). We show the boundary conditions for co-creation to be mutually beneficial and find that there is both a minimum and maximum level of manufacturing flexibility, that there is a minimum demand needed, and that the co-created product quality is limited by both the customer and the firm. Second, we discuss the co-created product quality and the firm's operational flexibility levels as an outcome of the co-creation process, and discuss the business settings where the likelihood for a mutually beneficial co-creation increases. We find that co-creation thrives for initiatives in environments in which customers are sensitive to quality and not to price, customer demand for the product is high, and investment intensity is low. Moreover, we find that the co-created product quality is higher when the customer is the first to contribute to the co-creation, rather than the firm.

Third, we also discuss how sharing the innovation effort between the customer and the firm is an instrument to guide the co-creation process. We find that the customer's innovation share in the co-created product increases when the customer, rather than the firm, is first to contribute to the co-creation. This effect is further intensified in environments in which co-creation thrives (see second contribution).

Our model stems from a managerial need to explain co-creation success or failure and to identify when co-creation is beneficial for the customer and firm. Our insights help (1) explain and predict the co-creation success or failure of different co-creation initiatives (e.g., see the discussion in the "Managerial Insights" section), (2) advise companies in which environments co-creation initiatives are best launched, and (3) suggest how innovation efforts in the cocreated product should be shared. Overall, our intention is to help companies and their customers gain insights into their co-creation process and to offer an analytical co-creation model on which future co-creation research can build.

Relevant Literature

Our work mainly contributes to the streams of literature on co-creation, mass customization, flexibility, and (joint) new product development, in which research is scattered across the fields of management, marketing, innovation, and operations.

Co-creation has received increasing attention from researchers and practitioners over the past years (see the reviews of [START_REF] Bendapudi | Psychological implications of customer participation in co-production[END_REF], [START_REF] Alves | Value co-creation: Concept and contexts of application and study[END_REF][START_REF] Gemser | Co-creation with customers: An evolving innovation research field[END_REF][START_REF] Gemser | Co-creation with customers: An evolving innovation research field[END_REF]). Following the service dominant logic, co-creation implies that customers are not just recipients of a good, but actively engage in the customer-supplier relationship and innovation process, and are actual co-creators of value [START_REF] Grönroos | Value co-creation in service logic: A critical analysis[END_REF][START_REF] Lusch | Service-dominant logic: reactions, reflections and refinements[END_REF][START_REF] Payne | Managing the co-creation of value[END_REF][START_REF] Prahalad | Co-opting customer competence[END_REF][START_REF] Vargo | Evolving to a new dominant logic for marketing[END_REF].

Value co-creation can refer to different types of value (e.g., monetary gains or social benefits for the firm or customer), different processes (e.g., co-creating firms or a firm co-creating with its customer), and different actors (e.g., lead users, a crowd of customers, communities, ...) [START_REF] Franke | Complementing mass customization toolkits with user communities: How peer input improves customer self-design[END_REF][START_REF] Jeppesen | User toolkits for innovation: Consumers support each other[END_REF][START_REF] Saarijärvi | Value co-creation: theoretical approaches and practical implications[END_REF]. In user innovation literature, lead users are well-known to have a higher competence and engagement to co-create, and a larger potential to develop more innovative and radically new products [START_REF] Hamdi-Kidar | Comparing lead users to emergent-nature consumers as sources of innovation at early stages of new product development[END_REF][START_REF] Vernette | Co-creation with consumers: Who has the competence and wants to cooperate[END_REF]. Moreover, some studies have also considered the impact of the network organization and externalities. For instance, [START_REF] Syam | Co-creation with production externalities[END_REF] find that a centralized pattern of externalities among customers (with an expert lead user in the center) positively affects firm co-creation profit, and [START_REF] Baldwin | How user innovations become commercial products: A theoretical investigation and case study[END_REF] consider a redundancy reduction thanks to communities.

All these studies have contributed to our knowledge on co-creation, however, there is still a lot to learn on how firms and customers engage in the co-creation process and manage the co-creation of value [START_REF] Payne | Managing the co-creation of value[END_REF][START_REF] Ulaga | Capturing value creation in business relationships: A customer perspective[END_REF].

While majority of co-creation studies emphasize the benefits of co-creation [START_REF] Alves | Value co-creation: Concept and contexts of application and study[END_REF][START_REF] Gemser | Co-creation with customers: An evolving innovation research field[END_REF], they ignore situations when co-creation is not beneficial, a gap we aim to fill. The few studies that do consider downsides of co-creation mainly focus on the costs related to finding and interacting with customers (e.g., by setting up information systems to communicate with customers) [START_REF] Franke | Value creation by toolkits for user innovation and design: The case of the watch market[END_REF][START_REF] Mahr | The value of customer cocreated knowledge during the innovation process[END_REF][START_REF] Syam | Co-creation with production externalities[END_REF], the risk of disappointed customers if the co-designed product fails to meet their expectations [START_REF] Heidenreich | The dark side of customer co-creation: exploring the consequences of failed co-created services[END_REF], or value co-destruction [START_REF] Plé | Not always co-creation: introducing interactional co-destruction of value in service-dominant logic[END_REF].

In this article, we develop an analytical model (thereby answering the call of [START_REF] Syam | Co-creation with production externalities[END_REF] for more economic co-creation models) to assess the conditions that drive cocreation to be economically beneficial for either the collaborating customer or the firm or both.

While certain studies consider 'subjective and social' co-creation value (such as an increased customer status, joy, accomplishment, social esteem and knowledge, or firm reputation) [START_REF] Franke | Why customers value self-designed products: The importance of process effort and enjoyment[END_REF][START_REF] Fuchs | Customer empowerment in new product development[END_REF][START_REF] Nambisan | Virtual customer environments: testing a model of voluntary participation in value co-creation activities[END_REF], or consider value as the customer's value-in-use related to the customer's longitudinal experience during consumption and usage [START_REF] Grönroos | Critical service logic: making sense of value creation and co-creation[END_REF][START_REF] Ranjan | Value co-creation: concept and measurement[END_REF]; our analytical model compels the focus on (modelable and measurable) economic gains of the customer and firm (measured through consumer surplus and profit respectively). The focus on economic benefits is in line with the work of, for instance, [START_REF] Ueda | Toward value co-creation in manufacturing and servicing[END_REF], [START_REF] Grönroos | Value co-creation in service logic: A critical analysis[END_REF], [START_REF] Baldwin | How user innovations become commercial products: A theoretical investigation and case study[END_REF] and [START_REF] Syam | Co-creation with production externalities[END_REF]. [START_REF] Ueda | Toward value co-creation in manufacturing and servicing[END_REF] present a value-creation model, including the producer's profit and the consumers' utility gain, to study provided, adapted and co-creative value; [START_REF] Grönroos | Value co-creation in service logic: A critical analysis[END_REF] develops a framework to measure mutually created value by considering jointly created productivity gains; [START_REF] Baldwin | How user innovations become commercial products: A theoretical investigation and case study[END_REF] model the pathway of how user innovations become commercialized products, using consumer surplus and firm profit; and [START_REF] Syam | Co-creation with production externalities[END_REF] study the effect of production externalities on co-creation depending on the network organization, also including consumer surplus and firm profit in their model.

Although flexibility can play a primary role in co-creation [START_REF] Syam | Co-creation with production externalities[END_REF]Zhang et al., 2011), to the best of our knowledge, we are the first to model the relationship between a firm's manufacturing flexibility decision and the success of co-creation analytically. This operational lens contributes to current co-creation literature, which has mainly taken a marketing or innovation perspective [START_REF] Gemser | Co-creation with customers: An evolving innovation research field[END_REF].

A vast body of literature has examined operational flexibility (for a review, see [START_REF] Yu | Operational flexibility: Review and meta-analysis[END_REF]), defined as the ability to change or react with little penalty in time, effort, costs, or performance [START_REF] Upton | The management of manufacturing flexibility[END_REF] and to deal with risk [START_REF] Sreedevi | Uncertainty and supply chain risk: The moderating role of supply chain flexibility in risk mitigation[END_REF]. We argue that a firm's manufacturing flexibility is driven by the fixed investments therein and that it leads to an easier (i.e., less costly) derivation of a new product in production. This follows a similar logic to that of [START_REF] Van Den Broeke | Platform flexibility strategies: R&D investment versus production customization tradeoff[END_REF], who argue that flexibility (in platforms) increases fixed investments but reduces the unit cost to obtain a product. That study fits in the broader literature stream of product development (for a review of that literature, see [START_REF] Krishnan | Product development decisions: A review of the literature[END_REF]).

We draw inspiration from the product development literature to model the different costs involved in product design. For example, similar to the innovation sharing model of [START_REF] Bhaskaran | Effort, revenue, and cost sharing mechanisms for collaborative new product development[END_REF] and the platform evaluation model of [START_REF] Krishnan | Appropriateness and impact of platform-based product development[END_REF], we consider product quality as an indicator of innovation. In our model, we distinguish between the consumer surplus and the firm profit, something that is also common in the literature on collaborative product development, R&D collaborations, and investment sharing. For example, [START_REF] Bhaskaran | Effort, revenue, and cost sharing mechanisms for collaborative new product development[END_REF] examine investment, revenue, and innovation sharing as inter-firm mechanisms to manage risk and uncertainty in joint new product development (e.g., in a B2C environment in which a biotech and pharmaceutical company work together). While some studies focus on the impact of contractual structures on the R&D partnership's success [START_REF] Bhattacharya | A comparison of milestonebased and buyout options contracts for coordinating R&D partnerships[END_REF][START_REF] Savva | Opt-out options in new product co-development partnerships[END_REF], in our analysis, we examine how sharing development efforts between the customer and the firm serves as an instrument to guide the co-creation success.

The Model

In this section, we develop an analytical B2C model with recourse to a firm (f ) collaborating with its customer (c) for the development of a new product. We assume a development collaboration with only one customer (or a group of customers with similar characteristics, so that they appear as one customer), a monopolistic firm (i.e., we do not consider competition), and assume that both the customer and the firm are risk neutral. In our main model, to avoid unnecessary complexity, we do not explicitly consider the network context, a simplification also applied in the work of [START_REF] Grönroos | Value co-creation in service logic: A critical analysis[END_REF] and [START_REF] Grönroos | Critical service logic: making sense of value creation and co-creation[END_REF]. However, in Appendix, we do present an extension of our main model in case other customers buy the co-created product (seen as a positive externality), or multiple customers contribute to the co-creation.

Our model includes the consumer surplus (CS) and the profit of the firm (Π f ) to assess the benefits from co-creation for the customer and the firm.

First, the consumer surplus is positively driven by the difference between its willingness to pay and the price for the product, and its product demand. Demand D is modeled endogenously, using the demand curve D = α -βP + γθ (based on [START_REF] Banker | Quality and competition[END_REF]), with α the intrinsic demand potential for the firm, β the sensitivity to the unit product price P , and γ the sensitivity to the product quality θ. Derived from the demand curve, the value for the customer obtained from buying the product is given by the consumer surplus (see [START_REF] Hicks | The foundations of welfare economics[END_REF] and [START_REF] Moorthy | Market segmentation, self-selection, and product line design[END_REF]), and in our model equals α+γθ β -P * D 2 . The firm's revenues from selling the product increases with a higher demand and selling price, and is simply given by P * D.

Second, the customer and the firm each decide on their effort in the new product development (i.e., their innovation share in the co-created product innovation); represented by θ c and θ f , respectively. For reasons of simplicity and ease of interpretation, in our main model, we do not consider a difference in quality appreciation γ between the innovation contribution of the firm θ f and the customer θ c , and assume γ to be positive. In Appendix, we present an extension of our main model where we allow the customer to have a different quality sensitivity for the firm's and customer's innovation effort, denoted by γ f and γ c respectively, leading to the following demand D = α -βP + γ c θ c + γ f θ f . While some studies have argued a higher quality appreciation for products that are self-or user-designed (i.e., γ c > γ f ) [START_REF] Franke | Why customers value self-designed products: The importance of process effort and enjoyment[END_REF][START_REF] Schreier | The innovation effect of user design: Exploring consumers' innovation perceptions of firms selling products designed by users[END_REF], others have argued that for certain product categories, such as luxury fashion brands, there is a negative effect of user-design (i.e., γ f > γ c , where γ f can be higher than for mainstream fashion brands and γ c can even be negative) [START_REF] Fuchs | All that is users might not be gold: How labeling products as user designed backfires in the context of luxury fashion brands[END_REF].

In our analysis, we consider three scenarios: one where the customer and the firm simultaneously decide their development effort, and the two scenarios where either the customer or the firm is first to determine its development effort (i.e., sequential decision making). The customer's and firm's development effort θ c and θ f affect the final co-creation outcome, that is, the product quality θ, with θ = θ c + θ f . The customer's and firm's share in the final co-created product are given by 1 -λ and λ respectively, with θ c = (1 -λ)θ, θ f = λθ, and λ ∈ [0, 1]. Note that in product development literature, product quality is commonly used to represent the level of innovation and the products' functionalities and features (see [START_REF] Tatikonda | Integrating operations and marketing perspectives of product innovation: The influence of organizational process factors and capabilities on development performance[END_REF], [START_REF] Bhaskaran | Effort, revenue, and cost sharing mechanisms for collaborative new product development[END_REF] and [START_REF] Heese | Product line design with component commonality and cost-reduction effort[END_REF]).

In line with the innovation sharing model of [START_REF] Bhaskaran | Effort, revenue, and cost sharing mechanisms for collaborative new product development[END_REF] For simplicity, and without loss of generality, we do not model interaction costs (see [START_REF] Bhaskaran | Effort, revenue, and cost sharing mechanisms for collaborative new product development[END_REF] who model an interaction cost that is increasing in the level of innovation undertaken by both parties). This assumption implicitly creates a benefit of co-creation in our model, since our total development costs, given by

I d θ 2 f + I d θ 2 c = I d λ 2 θ 2 + I d (1 -λ) 2 θ 2 = I d θ 2 (2λ 2 + 1 -2λ)
, are always equal or smaller than the development costs if either the customer or firm develops the product alone, given by I d θ 2 (since 2λ 2 +1-2λ with λ ∈ [0, 1] is always smaller than 1). Including interaction costs, analogous to Bhaskaran and [START_REF] Bhaskaran | Effort, revenue, and cost sharing mechanisms for collaborative new product development[END_REF], would change the total development costs to (

I d + K)(θ 2 f + θ 2 c ) = (I d + K)θ 2 (2λ 2 -2λ + 1), which is not necessarily smaller than I d θ 2 if K is sufficiently high.
The co-creation benefit we assume is realistic because of collaboration and specialization benefits, or because of economies of integration [START_REF] Piller | Does mass customization pay? an economic approach to evaluate customer integration[END_REF].

While the co-creation benefit for the customer is driven by the valuation of the product, and the purchasing and development costs, the firm's profit is driven by revenues and development costs, but also by the production cost and fixed operational investment, both of which depend on the firm's manufacturing flexibility decision. As we assume that production occurs in-house, the production cost and investment in operations are not shared with the customer.

Achieving a higher level of flexibility requires that the firm invests more up front (e.g., in its facilities). Given the natural assumption of decreasing marginal returns from investments [START_REF] Van Mieghem | Risk mitigation in newsvendor networks: Resource diversification, flexibility, sharing, and hedging[END_REF], the firm's investment in flexibility equals Ip Cp , where I p indicates the investment parameter for operational flexibility and a lower C p indicates a higher level of flexibility (i.e., the smaller C p , the higher is the level of flexibility, and the faster is the increase of investment therein). We assume that the firm's unit production cost C p decreases as the firm's level of operational flexibility increases. In a similar vein, [START_REF] Van Den Broeke | Platform flexibility strategies: R&D investment versus production customization tradeoff[END_REF] assume that a higher level of (platform) flexibility increases fixed investment, but lowers the variable production cost to obtain a product. The firm's production cost is simply given by

C p * D.
As a result, the consumer surplus (CS) and the profit of the firm (Π f ) are given by

CS = α + γθ β -P D 2 -I d θ 2 c , (1)
Π f = (P -C p ) D - I p C p -I d θ 2 f . (2)
The sequence of decisions in the co-creation process between the customer and the firm is shown in Figure 1. At first, the firm decides its level of manufacturing flexibility C p . This is a long term decision, which is more difficult to change. The firm and customer then cocreate (i.e., share development efforts θ c and θ f in the final product θ), either simultaneously (scenario 1) or sequentially (scenario 2 and 3). Subsequently the firm decides price P and the customer responds with its product demand D.

The customer and the firm make their decisions (D, P, θ c , θ f and C p ) to maximize their benefit (i.e., CS and Π f respectively). Note that we assume that the customer and the firm will only engage in co-creation when CS and Π f are positive.

Filling in the demand curve D = α -βP + γθ, (1) and (2) become:

CS = α + γθ β -P (α -βP + γθ) 2 -I d θ 2 c , (3)
Π f = (P -C p ) (α -βP + γθ) - I p C p -I d θ 2 f . (4)
The first-order condition characterizing price that maximizes firm profit (4) is

∂Π f ∂P = α -2βP + γθ + βC p = 0. Since ∂ 2 Π f ∂P 2 = -2β < 0, the profit function is concave in price.
This leads to the following price decision by the monopolistic firm:

P * = α + γθ + βC p 2β . ( 5
)
The corresponding CS and Π f , after inserting P * back into (3) and ( 4) are:

CS = 1 8β (α + γθ -βC p ) 2 -I d θ 2 c , (6)
Π f = 1 4β (α + γθ -βC p ) 2 - I p C p -I d θ 2 f . (7)
In the next section we assess under which conditions co-creation is mutually beneficial and determine the co-created product quality, flexibility, and innovation share of the customer and the firm under the three scenarios discussed in Figure 1.

Analysis

Boundaries for mutually beneficial Co-creation

From the expressions ( 6) and ( 7), we can derive the boundary levels of the jointly co-created product quality θ in function of the firm's flexibility C p so that both the customer c and firm f economically benefit (i.e., CS and Π f ≥ 0):

Following the proof in Appendix, we propose:

Proposition 1. Co-creation is beneficial for both the customer and the firm when θ ∈

[θ c , θ c ] ∩ [θ f , θ f ];
otherwise either the customer or the firm or both want to "escape" co-creation rather than engaging in it. We refer to expressions (19) and (20) in Appendix for the calculations of these boundaries.

Proposition 1 confirms that co-creation does not always lead to a positive value for both the firm and the customer. It also follows that

Corollary 1. There is a minimum and maximum level of flexibility (C p , C p ) for mutually beneficial co-creation.

Corollary 2. There is a minimum level of demand (α) for mutually beneficial co-creation.

Corollary 3. Both the customer and firm limit the co-created product quality (θ c and θ f ).

Note that the condition in Corollary 2 (see expression ( 21)) becomes more stringent for higher price sensitivity β, lower quality sensitivity γ, and higher investment in product development and flexibility (I d and I p ). This can be an important insight in practice because it means that for firms that deal with customers that are sensitive to price rather than to quality (e.g., firms focusing on a low-end rather than a high-end customer segment) or for firms that are in an investment intensive environment in terms of development and manufacturing (e.g., companies in electronics), customer co-creation is less likely to be mutually profitable, unless the co-created product can elicit higher demand. In extreme cases, if the base demand for a product α is lower, and if the users negatively appreciate co-development efforts of other users (e.g., luxury fashion brands), involving customers in product development might not be optimal (as illustrated by [START_REF] Fuchs | All that is users might not be gold: How labeling products as user designed backfires in the context of luxury fashion brands[END_REF]).

In Figure 2, we visualize the intervals of Proposition 1 in the "product quality-flexibility " space, in which higher θ refers to a higher quality of the co-created product, and lower C p refers to higher (operational) firm flexibility. We show the region Ω b where both the surplus of the customer and the profit of the firm are positive (leading to an incentive for both to engage in co-creation), the regions where either the customer or the firm obtains a negative value and therefore want to escape co-creation (resp. Ω f and Ω c ), and regions where neither partner has a benefit and both want to escape co-creation (i.e., regions Ω n ). We treat the size of Ω b as the likelihood of co-creation: the larger the region, the higher is the probability that co-creation will occur. Note that (looking at Figure 3), similar to our discussion earlier on the requirements of α, we find that mutually beneficial co-creation is more likely (i.e., the size of Ω b increases) in quality sensitive environments with low investment intensity.

In Figure 2 we also indicate the final levels of flexibility and product quality (C * p and θ * ) resulting from the co-creation. Note that, following Figure 1, there exists three different results of (C * p , θ * ), but that for the setting in Figure 2

Levels of Product Quality and Operational Flexibility

Before deciding on the price, both the customer and the firm take part in creating the new product (see Figure 1). Depending on the type of co-creation approach, the customer and firm simultaneously determine their product development input θ c and θ f , or either of the two parties initiates the co-creation, followed by the other getting involved in the co-creation.

For example, for the IT-platform-enabled crowdsourcing example of Lego Ideas (Schlagwein and Bjorn-Andersen, 2014) and for the user-innovation example of Rodeo Kayaking [START_REF] Baldwin | How user innovations become commercial products: A theoretical investigation and case study[END_REF], the customer is first to develop part of the product, while at MiAddidas, the firm takes the first step in the (R&D) product development. Remark that Lego is first to invest in the platform where product ideas can be suggested, but that we consider a real change in the product design to determine innovation effort, hence used to determine which party is first to invest in the co-creation (i.e., which scenario we consider in Figure 1). At MiAddidas, the firm makes the first step in product development, by developing different configuration and customization possibilities for the customer.

We find that (for the proof, see the Appendix):

Proposition 2. Under the three different scenarios in Figure 1, the total created product quality is: Scenario 1:

θ * 1 = 3γ(α -βC p ) 8βI d -3γ 2 , (8)
Scenario 2:

θ * 2 = γ(24βI d -γ 2 ) (α -βC p ) (8βI d -γ 2 ) 2 -16βI d γ 2 , ( 9
)
Scenario 3:

θ * 3 = γ (6βI d -γ 2 ) (α -βC p ) (4βI d -γ 2 ) 2 -2βI d γ 2 . ( 10
)
Comparing the product qualities under the different scenarios, it always holds that:

θ * 1 < θ * 2 < θ * 3 . (11)
We can thus state that:

Corollary 4. The co-creation process where the customer is first to contribute to the product development (i.e., scenario 3) leads to the highest co-created product quality θ.

From Proposition (2), we also find that:

Corollary 5. The co-created product quality θ increases with higher flexibility (i.e., lower C p ).

We now discuss the optimal level of C p (for the proof, see Appendix):

Proposition 3. Under the three different scenarios in Figure 1, the flexibility level C p is:

(12 )

C * p = 1 2β -1 + i √ 3 α 3 27 - I p β 4z a - I p β 4z a - 2α 3 27 + I p β 4z a 1 3 + 1 2β -1 -i √ 3 α 3 27 - I p β 4z a + I p β 4z a - 2α 3 27 + I p β 4z a 1 3 + α 3β ,
with z a for the different scenarios of the co-creation process given by: Scenario 1:

z a = 16βI 2 d -4I d γ 2 (8βI d -3γ 2 ) 2 , Scenario 2: z a = 16βI 2 d (8βI d -γ 2 ) 2 -16γ 2 βI d ,
Scenario 3:

z a = I d (4βI d -γ 2 ) 3 (4βI d -γ 2 ) 2 -2βγ 2 I d 2 .
From Proposition (2) and (3), and based on partial derivatives (see the discussion in Appendix), we find that:

Corollary 6. The co-created product quality and flexibility increase (i.e., θ increases and This implies that the co-creation outcome θ is the highest when companies co-create with quality-sensitive customers with high product demand, who have a low sensitivity to price, and when they are in a low-investment-intensive environment (independent of the scenario of how the co-creation process takes place).

Sharing Product Development Effort to Guide Co-creation

An important instrument in the co-creation process is the distribution of the innovation effort in the co-created product. The different scenarios (see Figure 1) lead to different ways to share innovation efforts. Since

λ 1 = 2 3 , ( 13
)
Scenario 2:

λ 2 = 16βI d 24βI d -γ 2 , ( 14
)
Scenario 3:

λ 3 = 4βI d -γ 2 6βI d -γ 2 . ( 15
)
This shows that a change of α or I p does not affect the optimal innovation share of the customer and firm. It follows that when the customer is the first to contribute to the co-creation (i.e., scenario 3), the firm will have a smaller share in the co-created product compared to when simultaneously contributing to the co-creation (i.e., scenario 1) (since λ 3 < λ 1 ). We also find that, when the firm is the first to contribute to the co-creation (i.e., scenario 2), the firm will have a larger share in the co-created product compared to when simultaneously contributing to the co-creation (i.e., scenario 1) (since λ 2 > λ 1 ).

Hence, it follows that:

Corollary 8. The customer's share in the co-created product is the highest when the customer is the first to contribute to the co-creation (i.e., scenario 3), while the firm's share is the highest when it is the first to contribute to the co-creation (i.e., scenario 2) (since it holds

that λ 3 < λ 1 < λ 2 ).
This is an important insight for companies. It shows that to get customers involved in the co-creation and let them increase their development efforts, it is better to let them first make development efforts, and thus involve them in the early stages of the co-creation process.

Moreover, it positively impacts the co-creation outcome (see Corollary 4). This effect gets even stronger when dealing with quality-sensitive rather than price-sensitive customers (e.g., in a high-end segment) or when being in a low-investment-intensive environment in terms of investments in product development. The latter is based on the following Corrollary (derived from Proposition 4):

Corollary 9. The share in the co-creation of the customer and the firm when the customer and the firm initiate the co-creation respectively (i.e., 1 -λ 3 and λ 2 respectively) increases when the quality sensitivity γ increases, or the price sensitivity β and the product development investment I d decrease.

We refer to Appendix for an extension of these results in case γ c = γ f , I c d = I f d , and the product is sold to other users, or developed by a community of users.

Managerial Insights and Conclusion

Literature and business examples have mainly focused on the benefits of customer co-creation for both the customer and the firm. However, in this article we develop an analytical model that shows that either the firm or the customer or both might want to escape rather than engage in co-creation (see Proposition 1). We show (1) the conditions under which cocreation is mutually beneficial, (2) which conditions stimulate co-creation, and (3) how co-creation efforts should be shared. Table 1 (based on our propositions and corollaries) gives an overview of the impact of different parameters on the co-created product quality level, the firm's operational flexibility, the sharing of development effort, and the likelihood

, θ 2 , θ 3 1 Cp 1 , 1 Cp 2 , 1 Cp 3 λ 2 λ 3 Ω b Customer's demand α ↑ ↑ - - ↑ Customer's quality sensitivity γ ↑ ↑ ↑ ↑ ↑ Customer's price sensitivity β ↓ ↓ ↓ ↑ ↓ Investment in operational flexibility I p ↓ ↓ - - ↓ Investment in product development I d ↓ ↓ ↓ ↑ ↓
of mutually profitable co-creation (i.e., the size of Ω b ), under the three co-creation process scenarios in Figure 1 (denoted by subscripts 1, 2 and 3 respectively).

A key finding of our study is that co-creation is not always beneficial for both the customer and the firm and that B2C co-creation seems to thrive (i.e., has a higher likelihood of being mutually beneficial with an increase of Ω b and θ) in environments where the customer is quality sensitive rather than price sensitive, demand is high, and the investment intensity is low. Note that results might differ in business-to-business settings. There we found many examples of co-development (e.g., in high-tech and pharmaceutical industries) with the goal of sharing the high investment costs. Knowing the conditions under which co-creation is stimulated can help managers better target their co-creation initiatives. For instance, they can co-create together with their high-end rather than low-end customer segment and for products that require less development effort, and have a higher demand.

The extra analyses and extensions to our main model (provided in Appendix), further help to interpret our results. We find that when the innovation contribution of the firm (θ f ) is increasingly appreciated over the innovation contribution of the customer (θ c ) (i.e., η decreases, given γ c = ηγ f ), the optimal innovation share of the firm λ increases. In the extreme case, this can lead to co-creation no longer being optimal (i.e., λ = 1). This could explain why user-based design backfires in the context of luxury fashion brands (although they are high-end customers), where the user's input might be negatively perceived, leading to lower quality and a lower signal of high status [START_REF] Fuchs | All that is users might not be gold: How labeling products as user designed backfires in the context of luxury fashion brands[END_REF]. Moreover, a decrease in total development costs, for instance thanks to a lower development cost for customers (i.e., lower I c d ) will increase the co-created product quality and co-creation likelihood. This decrease in development costs can come from involving lead users or communities of innovators in co-creation, as this might lead to less development effort and less search redundancy [START_REF] Baldwin | How user innovations become commercial products: A theoretical investigation and case study[END_REF][START_REF] Hamdi-Kidar | Comparing lead users to emergent-nature consumers as sources of innovation at early stages of new product development[END_REF]. Although studied in a co-production setting (where the customer participates in the production process), [START_REF] Haumann | Engaging customers in coproduction processes: How value-enhancing and intensity-reducing communication strategies mitigate the negative effects of coproduction intensity[END_REF] also found that the effort and time investment required by the customer should not be too high, or it might negatively impact the customer's satisfaction. In Appendix, we also consider having multiple customers buying the product or being involved in the co-creation process.

This can have an impact on our main model's results. For instance, under scenario 1 (i.e., when the customer(s) and the firm simultaneously decide their innovation effort), having multiple customers increases the co-created product quality, but can lead to a larger share of the firm's innovation share when involving these customers in the co-creation process.

In the next paragraphs, we discuss several of our model findings and their managerial relevance using some business examples. We examine two textbook examples of co-creation initiatives at Lego and Adidas, and a co-creation project at a wood-processing company (see Table 2).

With Lego Ideas, a customer can submit an idea for a new Lego set by building a prototype, and if the idea receives more than 10,000 votes, it is potentially accepted for commercialization by Lego (Lego, 2018).

By contrast, MiAdidas, similar to NIKEiD, offers customers the possibility to design their personalized Adidas shoes online [START_REF] Piller | Mass customization at adidas: Three strategic capabilities to implement mass customization[END_REF]. Crucial for Adidas to co-create is its new, flexible manufacturing plant, which uses an automated digital manufacturing process (Financial Times, 2016). Note however, that in such a mass-customization environment, Wood company Interview with the company's supply chain manager flexibility is limited to a preset range of choices for the customer, and the customer is only involved in the last stage of development, after the firm (similar to scenario 2 in our analysis).

Although Adidas presented their flexible 'Speedfactories' as its future, only four years after opening these factories (end 2019), they announced they would shut them down [START_REF] Adidas | Adidas is shutting down the robotic factories that were supposed to be its future[END_REF]. In the end the high investments did not deliver the flexibility they hoped for, resulting in a limited number of models they were able to make and co-create with the customer. For

Adidas, the high investment to achieve the necessary flexibility (i.e., related to high I p ) were not worthwhile.

We also consider a wood-processing company, which is increasingly receiving specific customer requests (e.g., a tailored shelf for a closet). These requests are difficult for a company organized as a mass producer to handle. In the past, the company had occasionally produced customized products. However, the high cost of doing so was unprofitable for the firm. In 2018, the company bought an expensive laser-cutting machine (at almost e0.8 million), which allows it to cut all customized forms of wood, thus highly increasing its operational flexibility. A customer can now provide a desired product design, after which the wood-processing company transforms it into a digital design processed by the machine (i.e., product development effort is shared). Note that for this company, demand α per individualized product can be quite low and customers are quite price sensitive, reducing the potential for mutually beneficial co-creation.

These B2C co-creation examples show that successful co-creation goes hand in hand with a sufficiently flexible operations system at the firm, that is, a low C p . This can explain why in certain settings without a sufficient level of flexibility, co-creating a personalized product is difficult to achieve. Another finding is that customer demand should be sufficiently high to ensure mutually profitable co-creation. For example, the voting threshold at Lego helps ensure a minimum level of demand (α) and interest for the product. Other means for a company to ensure a minimum demand could be contractual agreements. The benefit of Lego's voting system is that it also stimulates customer engagement and product publicity.

In the successful co-creation examples, the customer also shares part of the development effort. In our study, we show that the level of sharing development effort can be a useful instrument in the co-creation process. From Proposition 1 we also show that the optimal level of the co-created product quality (i.e., the level of innovation) should not be too high.

This might explain the preset range of options for customization at Adidas or the design restrictions at Lego (e.g., the customer can only suggest ideas with a maximum of 3000 pieces, can only use existing molds, and cannot use licensed properties such as Star Wars, as this would increase development costs).

We also found that to get customers involved in the co-creation and let them increase their development efforts, it is better to let them first make development efforts. In practice, it means that, for example at LegoIdeas, where the customer is first to make development efforts, the firm will have to exert a lower share of the co-creation efforts than when it takes the first step in product development (e.g., such is the case at Miaddidas).

The successful co-creation examples were not necessarily successful from the start but oftentimes evolved into a success over time, proving that co-creation is a dynamic process (see [START_REF] Fuchs | Customer empowerment in new product development[END_REF]; [START_REF] Mahr | The value of customer cocreated knowledge during the innovation process[END_REF]). From a model perspective, we consider a co-creation project successful if, in the co-created product quality-flexibility space, it is situated in or evolved over time into region Ω b (i.e., the region where co-creation is mutually beneficial).

As shown with our results, our analytical model leads to interesting and new insights, and provides a general framework to discuss the co-creation process. Future research can further build on this model to include even more aspects of co-creation. Future research could examine how having multiple co-creating customers might affect co-creation benefits and when paying customers for a product design is beneficial (e.g., Threadless gives a cash reward for the design of a T-shirt). Moreover, herein we assume that there is no limitation on the communication mechanism for customers to express their product needs and that their sensitivity to quality and price is known, which is not always the case. Another fruitful avenue for research is to determine what happens when the customer or the firm is not always rational (i.e., not always profit driven), due to, for example, loyalty to a firm, power relations, or risk-averse behavior. Future research can also examine who gains most from the co-creation, and can take into account other gains than pure monetary ones (e.g., valuein-use for the customer). Last, future research should investigate co-creation over time (longitudinal studies), as we showed that it is an evolving process.

Proof of Proposition 2 (scenario 1)

The first-order conditions for the maximization of the consumer surplus CS (6) and the firm profit Π f (7) when simultaneously optimizing the innovation input of the customer and the firm (θ c and θ f ) are:

∂CS ∂θ c = γ 4β (α -βC p + γ (θ f + θ c )) -2I d θ c = 0 (22) ∂Π f ∂θ f = γ 2β (α -βC p + γ (θ f + θ c )) -2I d θ f = 0 (23)
This leads to:

θ c = (α -βC p ) γ + γ 2 θ f 8βI d -γ 2 , (24)
θ f = (α -βC p ) γ + γ 2 θ c 4βI d -γ 2 . ( 25
)
Simultaneously solving these, we find:

θ * c = γ (α -βC p ) (8βI d -3γ 2 ) , (26)
θ * f = 2γ (α -βC p ) (8βI d -3γ 2 ) . ( 27
)
Since θ = θ f + θ c , it follows that:

θ * = 3γ (α -βC p ) 8βI d -3γ 2 . ( 28
)
Proof of Proposition 2 (scenario 2)

In scenario 2, the customer determines θ c (given by ( 24)) after the firm has set its development share θ f . After filling in (24) in Π f (given by ( 7)),we get:

Π f = 16βI 2 d (8βI d -γ 2 ) 2 ((α -βC p ) + γθ f ) 2 - I p C p -I d θ 2 f . (29)
Based on the first order derivative, we find that:

θ * f = 16γβI d (α -βC p ) (8βI d -γ 2 ) 2 -16βI d γ 2 . ( 30
)
Inserting ( 30) into ( 24), it follows that θ c equals:

θ * c = γ (8βI d -γ 2 ) (α -βC p ) (8βI d -γ 2 ) 2 -16βI d γ 2 . ( 31
)
Since θ = θ f + θ c , it follows that:

θ * = γ (24βI d -γ 2 ) (α -βC p ) (8βI d -γ 2 ) 2 -16γ 2 βI d . ( 32
)
Proof of Proposition 2 (scenario 3)

In scenario 3, the firm determines θ f (given by ( 25)) after the customer has set its development share θ c . After filling in (25) in CS (given by ( 6)),we get:

CS = 2βI 2 d (4βI d -γ 2 ) 2 ((α -βC p ) + γθ c ) 2 -I d θ 2 c , (33)
Based on the first order derivative, we find:

θ * c = 2βγI d (α -βC p ) (4βI d -γ 2 ) 2 -2βγ 2 I d . (34)
Inserting ( 34) into (25), it follows that θ f equals:

θ * f = (4βI d -γ 2 ) γ (α -βC p ) (4βI d -γ 2 ) 2 -2βγ 2 I d . (35)
Since θ = θ f + θ c , it follows that:

θ * = γ (6βI d -γ 2 ) (α -βC p ) (4βI d -γ 2 ) 2 -2βγ 2 I d . ( 36
)
Proof of Proposition 3

To simplify the proof and to make it generalizable for all scenarios in Proposition (2) we introduce the notations z c , z f and z t . For example, for scenario 1 in Proposition (2), we simplify the expressions to

θ c = z c (α -βC p ), θ f = z f (α -βC p ), and θ = z t (α -βC p ), with z c = γ 8βI d -3γ 2 , z f = 2γ 8βI d -3γ 2 and z t = 3γ 8βI d -3γ 2 .
A similar reasoning can be applied for scenario 2 and 3. After deriving the optimal C p we will replace these constants by their actual value.

To determine the optimal C p , the firm looks at its profit (7), which can be expressed as:

Π f = (α -βC p ) 2 (1 + γz t ) 2 4β -I d z 2 f - I p C p . (37)
We replace (1+γzt) 2 4β -I d z 2 f by the notation z a .

Based on the first order derivative, it needs to hold that:

∂Π f ∂C p = -2βz a (α -βC p ) + I p C 2 p = 0 (38)
This means that it should hold that:

C 3 p - α β C 2 p + I p 2β 2 z a = 0 (39)
To obtain the depressed form of this cubic equation, following Cardano's formula, we replace C p by y + α 3β and the condition becomes:

y 3 - α 2 3β 2 y - 2α 3 27β 3 + I p 2β 2 z a = 0. (40)
Now, we replace 2α 3 27β 3 -Ip 2β 2 za by z b , and y by u + v, so that (40) becomes:

(41) (u + v) 3 - α 2 3β 2 (u + v) -z b = 0, or (42) u 3 + v 3 + 3uv - α 2 3β 2 (u + v) -z b = 0. Setting 3uv -α 2 3β 2 = 0 and u 3 + v 3 = z b , v = α 2 9uβ 2 and u 3 + α 2 9uβ 2 3 = z b . Transformed this becomes u 6 -z b u 3 + α 2 9β 2 3 = 0 This means that u 3 = z b ± z 2 b -4 α 2 9β 2 3 2
. Thus, it holds that:

u 3 = z b -z 2 b -4 α 2 9β 2 3 2 , ( 43
) v 3 = z b + z 2 b -4 α 2 9β 2 3 2 . ( 44
)
Replacing the values of u and v (based on ( 43) and ( 44)) in y = u + v, and inserting this value of y in C p = y + α 3β : the first real root solution for C p is:

C p 1 =     z b -z 2 b -4 α 2 9β 2 3 2     (1/3) +     z b + z 2 b -4 α 2 9β 2 3 2     (1/3) + α 3β (45)
.

When replacing z b this becomes:

(46)

C p 1 = 1 β α 3 27 - I p β 4z a - I p β 4z a - 2α 3 27 + I p β 4z a 1 3 + 1 β α 3 27 - I p β 4z a + I p β 4z a - 2α 3 27 + I p β 4z a 1 3 + α 3β . .
This result can also be easily obtained with the cubic formula. As a cubic equation always has three roots, the other roots are given by: (47)

C p 2 = 1 2β -1 + i √ 3 α 3 27 - I p β 4z a - I p β 4z a - 2α 3 27 + I p β 4z a 1 3 + 1 2β -1 -i √ 3 α 3 27 - I p β 4z a + I p β 4z a - 2α 3 27 + I p β 4z a 1 3 + α 3β . (48)
C p 3 = 1 2β -1 -i √ 3 α 3 27 - I p β 4z a - I p β 4z a - 2α 3 27 + I p β 4z a 1 3 + 1 2β -1 + i √ 3 α 3 27 - I p β 4z a + I p β 4z a - 2α 3 27 + I p β 4z a 1 3 + α 3β .
Note that complex numbers always come into play when applying the cubic formula. The discriminant of Equation (39) equals Ip The values of z a for the different scenarios of the co-creation process can be expressed as:

Scenario 1: (49) z a = 4I d [4βI d -γ 2 ] (8βI d -3γ 2 ) 2 , Scenario 2: (50) z a = 16βI 2 d (8βI d -γ 2 ) 2 -16γ 2 βI d , Scenario 3: (51) z a = I d (4βI d -γ 2 ) 3 (4βI d -γ 2 ) 2 -2βγ 2 I d 2 .
Impact of different parameters on C p and θ Replacing α 3β by x 1 and Ip 4β 2 za by x 2 , C * p becomes:

(52)

C * p = 1 2 -1 + i √ 3 x 3 1 -x 2 --x 2 (2x 3 1 -x 2 ) 1 3 - 1 2 1 + i √ 3 x 3 1 -x 2 + -x 2 (2x 3 1 -x 2 ) 1 3 + x 1 .
Given that x 3 1 > x 2 it holds that C p increases if x 1 decreases and x 2 increases.

A decrease of α and an increase of β lead to a decrease of x 1 , and hence an increase of

C p .
Under the different scenarios (see Figure 1), x 2 becomes (based on Proposition 3 for the values):

Scenario 1:

x 2 = Ip(8βI d -3γ 2 ) 2 16β 2 I d [4βI d -γ 2 ]
Scenario 2: It holds that in all scenarios, if I p increases, x 2 increases, and hence C p increases.

We show the impact of different parameters on x 2 (with partial derivatives) for scenario 1. A similar logic holds for scenario 2 and 3.

Note that x 2 decreases with an increase in β, so that C p should decrease. This leads to opposite findings compared to the impact of β on x 1 , and hence C p . However, the general impact of an increase of β is an increase of C p . Distinguishing between the quality sensitivity of the customer to the product innovation share of the customer and firm, demand changes to D = α -βP + γ c θ c + γ f θ f . Consequently, the consumer surplus and firm profit (Equations ( 6) and ( 7)) change to:

Given

CS = 1 8β (α -βC p + γ c θ c + γ f θ f ) 2 -I d θ 2 c , (55)
Π f = 1 4β (α -βC p + γ c θ c + γ f θ f ) 2 - I p C p -I d θ 2 f . ( 56
)
Analogous to the proof of Proposition 2, we find the following co-created product quality θ and the innovation share of the firm λ, based on Equations ( 55) and ( 56), under the three scenarios in Figure 1:

• Scenario 1: θ = (α-βCp)(γc+2γ f ) 8βI d -2γ 2 f -γ 2 c , λ = 2γ f γc+2γ f ,
• Scenario 2: θ = .

An increase of γ f always leads to an increase of θ if η = γc γ f > -2 and to a decrease if η = γc γ f < -2. For the example of luxury fashion brands, where γ c < γ f and γ c can be negative, this implies that optimally, the development share of the firm λ = 6) and ( 7)) change to:

Π f = (α + γθ -βC p ) 2 (N + 1) 4β - I p C p -I d θ 2 f . (60)
Analogous to the proof of Proposition 2, we find the following co-created product quality θ and the innovation share of the firm λ, under the three scenarios in Figure 1:

• Scenario 1: θ = 3γ(N +1)(α-βCp) 8βI d -3γ 2 (N +1) , λ = 2 3 ,

• Scenario 2: θ = (α-βCp)γ(24βI d -γ 2 (N +1)) (8βI d -γ 2 (N +1)) 2 -16βI d γ 2 (N +1) , λ = 16βI d 24βI d -γ 2 (N +1) ,

• Scenario 3: θ = (α-βCp)(N +1)γ(6βI d -γ 2 (N +1)) (4βI d -γ 2 (N +1)) 2 -2βI d γ 2 (N +1) , λ = 4βI d -γ 2 (N +1) 6βI d -(N +1)γ 2 .

Next, we consider the situation where the N neighbors join in the co-creation process:

The consumer surplus (CS) and firm product (Π f ) (Equations ( 6) and ( 7)) change to:

CS = (α + γθ -βC p ) 2 (N + 1) 8β -I d (N + 1) 2 θ 2 c , (61)
Π f = (α + γθ -βC p ) 2 (N + 1) 4β -

I p C p -I d θ 2 f . (62)
Analogous to the proof of Proposition 2, we find the following co-created product quality θ and the innovation share of the firm λ, under the three scenarios in Figure 1:

• Scenario 1: θ = γ(N +1)(2N +3)(α-βCp) 8βI d (N +1)-2γ 2 (N +1) 2 -γ 2 , λ = 2N +2 2N +3 ,

• Scenario 2: θ = (α-βCp)(N +1)γ(16βI d (N +1) 2 +8βI d (N +1)-γ 2 )

(8βI d (N +1)-γ 2 ) 2 -16βI d γ 2 (N +1) 3 , λ = 16βI d (N +1) 2 8βI d (N +1)-γ 2 +16βI d (N +1) 2 ,
• Scenario 3: θ = (α-βCp)γ(N +1)((N +1)(4βI d -(N +1)γ 2 )+2βI d )

(N +1)(4βI d -(N +1)γ 2 ) 2 -2βI d γ 2
, λ = (N +1)(4βI d -γ 2 (N +1)) 2βI d +(N +1)(4βI d -(N +1)γ 2 ) .

  , development costs in our model are given by the polynomial function I d θ 2 c and I d θ 2 f for the customer and the firm respectively, with I d equal to the investment parameter for product development (i.e., a higher I d means a more investment-intensive environment for product development). In Appendix, we also extend our main model and analyze what happens if the development cost for the customer would differ from that of the firm, denoted respectively by I c d and I f d . The development cost of the customer might decrease (and be lower than that of the firm, with I c d < I f d), e.g., when involving a lead user in the co-creation, rather than an ordinary customer[START_REF] Hamdi-Kidar | Comparing lead users to emergent-nature consumers as sources of innovation at early stages of new product development[END_REF].

Figure 1 :

 1 Figure 1: Decision timeline of the co-creation process.

Figure 2 :

 2 Figure2: The firm's flexibility and the co-created product quality determine whether cocreation is beneficial for both the customer and the firm (i.e., in the region Ω b ). We also indicate the optimal levels of flexibility and product quality (C * p , θ * ). The figure is constructed given β = 2, γ = 1, I d = 2, I p = 3, α = 6.7, and λ = 0.45.

Figure 3 :

 3 Figure 3: Panel (a) visualizes the same environment as in Figure 2, while panel (b) and (c) illustrate that an increase in quality sensitivity γ and an increase in product development investment I d , increase and decrease the size of Ω b (i.e., the region where co-creation is mutually beneficial) respectively.

C

  p decreases) with an increase of quality sensitivity γ and demand α, Corollary 7. The co-created product quality and flexibility decrease (i.e., θ decreases and C p increases) with higher investment in product development and flexibility (I d and I p ) and price sensitivity β.

  λ) , we can derive λ (based on the results in Appendix when proving Proposition (2)): Proposition 4. Under the three different scenarios the innovation share of the co-created product carried by the firm is: Scenario 1:

4z 2 a

 2 β 5 [8z a α 3 -27I p β]. As this discriminant is larger than zero, three real solutions exist for C p . From all suggested solutions for C p (i.e.,C p 1 , C p 2 , C p 3 ) only C p 2 ∈ [C p , C p ] (i.e., lies in the region Ω b in Figure2, hence why this is the optimal solution for C * p .)

  d (4βI d -γ 2 ) 3

,•

  (α-βCp)(8βI d (γc+2γ f )-γ 3 c ) (8βI d -γ 2 c ) 2 -16βI d γ 2 f , λ = 16βI d γ f 8βI d (γc+2γ f )-γ 3 c Scenario 3: θ = (α-βCp)(2βI d (γc+2γ f )-γ 3 f ) (4βI d -γ 2 f ) 2 -2βI d γ 2 c , λ = γ f (4βI d -γ 2 f ) 2βI d (γc+2γ f )-γ 3 f . Let us denote γc γ f = η. As an illustration, for scenario 1, ∂θ ∂γ f = (α-βCp)(η+2)(8βI d +γ 2 f (2+η 2 )) (8βI d -γ 2 f (2+η 2 )) 2

  increase, and that if γ c ≤ 0, involving the customer in the co-creation process does not create any value. Impact when I c d differs from I f d Distinguishing between the development cost of the customer and the firm, the consumer surplus and firm profit (Equations (

Table 1 :

 1 Impact of Different Parameters (Summary of Propositions and Corollaries) Increase of different parameters θ 1

Table 2 :

 2 Overview of the different cases

	Case Company Description Source of Information
	A	Lego	Existing literature
	B	Adidas	Existing literature
	C

  Zhang, Xiang, Chen Ye, Rongqiu Chen, Zhaohua Wang. 2011. Multi-focused strategy in value co-creation with customers: Examining cumulative development pattern with new capabilities. International Journal of Production Economics 132(1) 122-130.

  if γ increases, x 2 decreases, and hence C p decreases.Impact when γ c differs from γ f

	∂x 2 ∂γ	=	-2γI p (8βI d -3γ 2 ) 16β 2 I d	(16βI d -3γ 2 ) (4βI d -γ 2 ) 2 < 0,	(53)
	it follows that Given
	I p γ 2 (8βI d -3γ 2 )	16βI d -3γ 2
			16β 2	I 2 d [4βI

d -γ 2 ] 2 > 0, (

54

)

it follows that if I d increases, x 2 increases, and hence C p increases.

Appendix Proof of Proposition 1

From the model, it holds that θ = θ f + θ c , with θ f = λθ and θ c = (1 -λ)θ (with λ ∈ [0, 1]).

A positive consumer surplus (6) and firm profit (7) are then given by:

Based on the second derivative, both ( 16) and ( 17) are concave quadratic in θ, under the

This means it should also hold that:

Applying the quadratic formula, this results in the following limits, in terms of the total co-created product quality θ, for CS to be positive:

and the following limits for Π f to be positive:

For the discriminant in the above expression to be positive, it needs to hold that there is a minimum base demand α required for the product:

Analogous to the proof of Proposition 2, we find the following co-created product quality θ and the innovation share of the firm λ, based on Equations ( 57) and ( 58), for scenario 1 (note that scenario 2 and 3 follow a similar logic): θ =

Impact of multiple buyers or multiple co-creating customers

Let us consider a situation where there is a community/network of multiple customers. We consider a 'main' customer, who always participates in the co-creation with the firm, and N neighbors of this customer; the latter can either participate in the co-creation or not. Below, we try to give insights on the positive externalities in co-creation when being able to sell to or co-create the product with multiple customers. The final co-created product quality is

given by θ = (N + 1)θ c + θ f (with θ f = λθ and θ c = (1-λ)θ (N +1) ) when N + 1 customers join the co-creation process, or θ = θ c + θ f (with θ f = λθ and θ c = (1 -λ)θ) when they don't join the co-creation process. The total demand curve equals D = (α -βP + γθ)(N + 1). The latter assumes there is no price discrimination between the main customer and its neighbors for the product, and that they have the same sensitivity to price and quality. Note that if N = 0, we fall back to the main model analyzed in the core of the paper.

Let us first consider the situation where the N neighbors buy the product, but do not join the co-creation process:

The consumer surplus (CS) and firm product (Π f ) (Equations ( 6) and ( 7)) change to:


Lu Zhao

Xiaowen Zhu

Jonathan Dumas

Didier Chirpaz

Tom Joannem

Francis Gohier

Nicolas Solenne

Patrice Guiho

Remy Braud

Olivier Piquet

Guillaume Ferrand

Sébastien Ladegaillerie

Claude Marchand

Francoise Gougnaud

Romuald Duperrier

Didier Uriot

Leo Weissman

Boaz Kaizer

Eli Farber

Matthieu Michel

Jean-Francois Leyge

Study and test of the rebuncher for SARAF-LINAC phase II

Keywords: RF power, Buncher, MEBT, Beam dynamic, Linac

In order to match the beam exiting the RFQ to the superconducting cavities and to minimize the longitudinal beam extension, three rebunchers will be installed in the Medium Energy Beam Transport (MEBT) of SARAF-LINAC phase II.

CEA is in charge of the design and fabrication of the copper-plated stainless steel, 3-gap rebunchers. Beam dynamic simulation leads to the global MEBT design. The rebuncher was successfully conditioned to the RF power of 4.4 kW required for CW operation, with sufficient power margin. A solid state power amplifier of 10 kW developed by SNRC is used during the RF test. The rebuncher shows a good performance in terms of the dissipated power, peak temperatures and vacuum level.

Introduction

The upgrade of the Soreq Applied Research Accelerator Facility (SARAF), managed by the Soreq Nuclear Research Center (SNRC), aims to deliver 5 mA CW deuteron and proton beams at 40 and 35 MeV respectively [START_REF] Kreisel | Saraf phase-i beam operation status[END_REF]. The Commissariat à l'Energie Atomique et aux énergies alternatives (CEA) is in charge of the design and building of a Medium Energy Beam Transport (MEBT) line and the SuperConducting Linac (SCL) [START_REF] Pichoff | The SARAF-LINAC Project 2019 Status[END_REF] to complete the existing structure of phase I [START_REF] Berkovits | Operational experience and future goals of the saraf proton/deuteron linac[END_REF] consisting of an ECR ion source, a Low Energy Beam Transport (LEBT, a Radio Frequency Quadrupole (RFQ) and an High Energy Beam Transport (HEBT) line. Figure 1 illustrates a schematic drawing of the SARAF-LINAC project. The SCL has been discussed in the previous article [START_REF] Pichoff | Discussion on SARAF-LINAC Cryomodules[END_REF]. A 5-meter MEBT line (see Figure 2) consisting of three normal conducting rebunchers, eight quadrupoles, three H/V slit boxes, two diagnostics boxes and Structural-Vacuum-Cooling (SVC) sub-system, plays no acceleration goal but has other essential functionalities [START_REF] Pichoff | The saraflinac project for saraf-phase 2[END_REF]:

1. to match the beam from the RFQ to the SCL, 2. to measure the beam current and the transverse and longitudinal beam distributions, [START_REF] Berkovits | Operational experience and future goals of the saraf proton/deuteron linac[END_REF]. to steer the beam on the accelerator axis, 4. to clean the beam if necessary (elimination of halo) to stop the pulsed beam (low power) during the tuning process of the injector alone, 5. to cope with the high vacuum gradient between RFQ and downstream SC structure. We use the TraceWin code [START_REF] Uriot | Status of tracewin code[END_REF] to optimize the beam performances. The beam dynamic analysis defines the optimal position of different elements of the MEBT to manage the beam coming from RFQ and match it to the SCL. Three rebunchers are mandatory to avoid an excessive longitudinal extension of the beam which would cause unacceptable emittance growth and losses in SCL [START_REF] Dumas | Beam dynamic studies for the saraf mebt and sc linac[END_REF].

Different configurations of the bunching cavity, such as number of gaps, stem geometry, have been studied in order to minimize the total power consumption and the peak power density [START_REF] Michel | Saraf internal repport: Pdr of saraf-linac major components mebt rebuncher[END_REF]. The IFMIF-type copper-plated stainless steel cavity was selected to operate at 176 MHz. Numerical simulation was performed to optimize the RF design and the thermal budget of rebuncher. The specification of the rebuncher were defined and given in Table 1. To compensate the mechanical manufacturing tolerances, the simulation error and the thermal drift, a frequency tuning system is needed. A closed-loop control system drives the tuning system according to the measured phase between the injected and transmitted RF signals of the rebuncher. [START_REF] Leyge | Saraf internal report[END_REF]. In order to ensure RF impedance continuity, multi-contacts have to be installed at each flange besides the vacuum metal sealed joint.

To validate the rebuncher design, the RF power test for each rebuncher is essential before installing in the MEBT line. Furthermore, the RF power test can clean the inner side of rebuncher to enhance its vacuum performance.

Beam dynamic of MEBT

Beam dynamic studies define the specifications for the rebunchers. In the regular operation mode of the SARAF machine, the MEBT adapts the beam from the RFQ to the SC-Linac and needs to include an optional chopper, not yet defined but deviates the beam to a beam dump with a voltage of 800V, which makes the MEBT longer and calls for 3 rebunchers with max. E 0 TL of 120kV (see on-axis field of the rebunchers in Figure 3).

55

The most demanding power for the rebunchers is set by the highest intensity beam of the heaviest species considered (D + , 5mA). The Twiss parameters of this beam from the RFQ are shown in table 2. The goal is to keep the beam footprint in the longitudinal phase space at the 65 end of the MEBT within a 0.2 MeV and 50 • range around the reference particle in order to match the SC-Linac acceptance, see Figure 5. During the commissioning of the machine, the longitudinal emittance of the beam coming from the RFQ needs to be evaluated. One of the proposed methods is to measure the bunch length evolution when varying the effective voltage in the first rebuncher [START_REF] Strehl | Beam Instrumentation and Diagnostics[END_REF]. In these conditions, an effective voltage up to 150 kV is necessary as shown in Figure 6, in order to reach a minimum in the bunch length and derive the Twiss parameters. Nevertheless, this measurement will be done only with pulsed beam and with pulsed RF in the rebuncher, (duty cycle (dc) < 40%) to limit the rebuncher warming. A lower voltage would affect the precision of the beam longitudinal emittance evaluation leading to the degradation of the longitudinal beam matching to the SCL, such as emittance growth and higher beam loss rate.

RF study

The rebuncher design is based on the experience accumulated with the SPI-RAL2 rebuncher [START_REF] Di Giacomo | Design of the mebt rebunchers for the spiral2 driver[END_REF]. Considering the total power consumption and the peak power density within the allocated length of 280 mm flange to flange, the IFMIFtype cavity, declined in 3-gap version, with 40 mm aperture, fits well our requirements. The basic structure of the 3-gap rebuncher has been given in our previous published work [START_REF] Kaizer | Study and development of CW room temperature rebuncher for SARAF accelerator[END_REF]. Figure 7 gives a general view of SARAF2 rebuncher with the main components separated. Cu-plated stainless steel constitutes the outer tank. However, solid copper is used for the fabrication of quarter wave stems and the stem flanges, where most of the power is dissipated and where higher thermal conductivity is mandatory. Solid copper is also used for beam ports and drift tubes, which endure the high electric field as well. To keep the leakage rate better than several 10 -8 mbar l/s , the double gaskets have been preferred

for each flange connection: a vacuum Al joint with the GETELECT spira-shield commercial RF gaskets are used for the front flange. For other flange connections, the commercial CF copper gaskets (different sizes) are used with the same RF gaskets.

Tuner

Figure 8(a) shows the inner view of the rebuncher consisting of a fixed, a motorized tuners, two stems and a power coupler loop. Two tuner holders are welded on the tank bottom to limit the top flange deformation (in order to limit the frequency shift) under vacuum and to isolate higher order modes caused by the movable tuner. The height of cylinder (fixed tuner) has to be evaluated before copper plating process to have symmetrical field distribution around the stem rings. That is to say, the rebuncher should have the resonance frequency of 176 MHz under the required vacuum condition when the tuner distance (the distance between the movable tuner and the stem ring) equals to the fixed distance (the distance between the fixed tuner and the stem ring). The tuner can compensate the frequency deviation due to the mechanical deformation induced by power dissipation etc. The movable tuner swings slightly around the symmetrical position in the regular operation. Thus, the effect of non-symmetrical field can be neglected.

The movable tuner has a tuning range of 30 mm which corresponds to 1.5

MHz frequency tuning range as shown in Figure 8(b) (blue line). The measured frequency is higher than the simulated value (red line), because the frequency is measured under the atmospheric condition (1 bar), the frequency will decrease about 230 kHz due to the rebuncher deformation under the vacuum condition. Thus, considering this decrease in frequency (red plus plot), the simulated results show a good agreement to the measured values.

Frequency vs Temperature

Considering the initial cooling water temperature of T 0 = 25 • C, the rebuncher body temperature with full power energy is about 55 • (given in section 4). The temperature raise of 30 • C will induce a series of variations of the rebuncher, such as, the thermal expansion and the deformation at some hot spots of rebuncher. These will cause a frequency drift about of 98 kHz for resonant frequency of 176 MHz.

When the generator frequency f is very close to the cavity resonance frequency f 0 , the relation between the detuned phase angle φ and frequency detuning ∆f is given by the equation [START_REF] Schilcher | Vector sum control of pulsed accelerating fields in lorentz force detuned superconducting cavities[END_REF]:

tanφ ≈ 2Q L ∆f f (1)
So that one degree variation of temperature can yield the frequency tuning about ∆f = 3 kHz, corresponding a phase variation of 6.8 • calculated by the equation 1.

Thermal study

The main cavity and flanges are in stainless steel with copper coating inside.

This technology is less expensive than massive copper, less bulky and easier to weld. The difficulty is to cool properly stainless steel because of its low conductivity compared to copper (16 W/m/k against 400 W/m/K for copper).

Every heated area has to be cooled locally. A series of multiphysics coupled RF and thermal-stress analyses have been performed to investigate the temperature map of rebuncher. The simulation was run with the code ANSYS. Thermal calculations have been performed with thermal load increased by two margin coefficients: 20% on total RF loss, which were calculated with a 30% margin on the surface conductivity. Then we expect our results to be overestimated.

Moreover, they are obtained at design voltage (120 kV) and not at required voltage (105 kV) [START_REF] Michel | [END_REF]. According to our experiences, we designed the cooling circuit (c1 to c5 and c8, c9) grooved in the material thickness of the cavity, while two copper pipe brazed out of the caviy wall for C6 and C7, as shown in Figure 9. Inside the cavity, the stems made of copper are cooled by conduction from the stem neck and the movable tuner is cooled by the copper pipe brazed inside as shown in Figure 10.

RF test

RF performances of the rebuncher were measured using a Vector Network 155

Analyzer. The quality factor Q 0 of the rebuncher is about 7000 calculated by

Q 0 ≈ 2 × Q L where Q L is measured of 3500.

RF power amplifier

A compact narrow-band Solid State Power Amplifier(SSPA) capable of delivering 10 kW CW RF power at 176 MHz was developed at SNRC for SARAFlinac Phase II. The 10 kW amplifier is built by combining the RF power from two 5 kW amplifiers using 3 dB 90 o High Power Quad Hybrid Couplers (HPQHC).

The 5 kW amplifiers are combined from eight 800 W LDMOS water cooled amplifier modules equipped with an internal circulators and RF loads. The conceptual electrical diagram is presented in Figure 11. This RF amplifier is installed at the outside of the rebuncher RF test bunker. The amplifier was first tested by connecting the amplifier output to a water cooled 50 Ω load, and driven by an RF generator, it was tested up to maximum output power of 12kW. A 1'5/8 rigid coaxial line was used to conduct the RF power from the RF power amplifier to the rebuncher.

Acquisitions and Control system

Under the CEA EPICS environment (IEE framework [START_REF] Gougnaud | Cea irfu epics environment for the saraf-linac project[END_REF]), a control system for the test bench (data acquisition, system configuration and automatic conditioning) is developed using EPICS 3.15 [16]. The system runs on a IOxOS IFC1210 VME card, and performs the high speed acquisitions (RF measurements) through an IOxOS ADC3111 mezzanine card (8 channels ADC 16-bit 250 Mps). It performs as well the slow acquisitions and controls (analogue or binary) through remote I/O Beckhoff modules using Modbus protocol with an EK9000 coupler. Experiments measurements and controls are archived continuously using EPICS Archiver Appliance [START_REF] Shankar | The epics archiver appliance 280[END_REF].

Closed-loop tuning control

In the closed-loop tuning control system, a Siemens 1500 PLC and a S110 speed variator is responsible for driving the movable tuner based on the measured phase between the injected and transmitted RF signal. Figure 12 gives the response of the phase detector of AD8302 which shows a good linear in the range from -50 • to 50 • . The control of motor is divided in two regions: regu- lation and fast movement. For example, the phase between -20 • to 20 • (see figure 13) is the regulation area where the motor will move as a speed of 2 mm/s because of the relative low reflected power. Once the measured phase out of this area, the motor will bring the phase directly to 0 • with a higher speed in order to avoid too much reflected energy. Note that, the closed-loop control will become disabled once the injected RF signal stops. During the RF power test, a slight phase variation was observed. Figure 14 illustrates the measured phase variation due to the cooling water temperature variation of 1 • C. The 0.037 V

RF Power Test

Figure 15 shows the rebuncher RF power test bench inside the bunker with the radiation shielding. The rebuncher has been successfully conditioned up to pulse ratio of 1% with an RF pulse repetition of 1 Hz. With the control system, the power level and the pulse ratio increase automatically. The RF power will decrease in case of a vacuum burst, ex. excessive reflections or sparking. Vacuum in the rebuncher is maintained to be better than 1 × 10 -7 mbar during the RF conditioning. We observe that most multipacting occurs in the low energy region (from 200 to 550 Watt).

In addition, the gas composition in the rebuncher was changed by the RF conditioning as shown in table 5 and the vacuum quality was enhanced by a factor of three.

Conclusion

According to the beam dynamic analysis, the requirement and configuration of each element in the MEBT line have been defined. A copper-plated stainless steel, 3-gap IFMIF-type rebuncher design was selected. The numerical simulations allow us to investigate and optimize the initial rebuncher design.

The first fabricated rebuncher shows a stable and reliable performance during RF power tests (pulsed and CW mode). Our experiments conform in line with our previous studies. The RF conditioning enhances the vacuum level of a factor of three thus reaching a good vacuum level of 2.1 × 10 -8 mbar. The max. temperature of the rebuncher during the power test is much lower than the critical value so that the cooling system performed better than expected.

Facing the thermal (cooling water temperature variation and RF loss) induced frequency shift, one side motorized tuner with the closed-loop control system guaranties the rebuncher frequency stability. In addition, the RF power tests validate the copper-plating technology used by the fabrication company. It is useful to guide the two following rebunchers' fabrication and reduce the risk.

Nevertheless, the longitudinal emittance of the beam from the RFQ should be evaluated during the commissioning of the machine to validate all the rebuncher performances.

Acknowledgment

We would like to thank DI GIACOMO Marco for useful input and comments regarding preliminary design.

Figure 1 :

 1 Figure 1: Schematic view of SARAF-LINAC:

Figure 2 :

 2 Figure 2: MEBT line 3D model.

Figure 3 :

 3 Figure 3: On-axis field rebuncher for E 0 TL= 120 kV.

60 and beam envelope are illustrated in Figure 4 .

 4 Envelope peaks are optimized in order to minimize the biggest peak, inducing by this way the balancing of quadrupole and rebuncher forces, therefore minimizing also the biggest forces[START_REF] Dumas | Beam dynamic studies for the saraf mebt and sc linac[END_REF].

Figure 4 :

 4 Figure 4: 3σ transverse (left), x in blue and y in red. and longitudinal (right) envelopes in the MEBT. [7]

Figure 5 :

 5 Figure 5: MEBT longitudinal distribution (in red), compared with the SC-linac acceptance (in white).

Figure 6 :

 6 Figure 6: RMS beam bunch vs voltage in the first rebuncher.

Figure 7 : 3 -

 73 Figure 7: 3-gap rebuncher cavity components[START_REF] Kaizer | Study and development of CW room temperature rebuncher for SARAF accelerator[END_REF]

Figure 8 :

 8 Figure 8: Rebuncher with tuning system

Figure 9 :

 9 Figure 9: Cooling circuits of the cavity tank and zoom on the stem flange .

Figure 10 :

 10 Figure 10: Inner cooling circuits.

Figure 11 :

 11 Figure 11: 10kW HPA conceptual diagram.

Figure 12 :

 12 Figure 12: Output of phase detector.

Figure 13 :

 13 Figure 13: speed regulation for phase difference tuning

Figure 14 :

 14 Figure 14: Phase variation as water temperature.

Figure 15 :

 15 Figure 15: Full equipped rebuncher for RF power test.

200 4 .

 4 4 kW in CW mode at a frequency of 176 MHz. The RF conditioning starts at a gentle condition, for example, a very low power level of 10 W and an RF

Table 1 :

 1 rebuncher requirements

	Requirement	Value	Units
	Frequency	176	MHz
	Synchronous phase	-90	degree
	Quality factor	> 6600
	Drift tube aperture	40	mm
	Flange to flange distance < 280	mm
	Tuning range	> 600	kHz
	Max. local temperature	100	• C
	Additionally, some technical conceptions have to be considered. According
	to our experiences from SPIRAL2, the rebuncher has to be properly cleaned to
	present residual outgassing (lower than 3 • 10 -5 mbar • l • s -1 ) when operated
	in CW, at design voltage. It should be equipped with a pumping CF flange,
	leaving an open surface higher than 45•cm 2

Table 2 :

 2 Twiss parameters for 5 mA deuton RFQ output

	α x	-0.56
	α y	0.06
	α z	-0.26
	β x 0.16 mm/π.mrad
	β y 0.20 mm/π.mrad
	β z 0.64 mm/π.mrad
	Norm. RMS xx	0.20 π.mm.mrad
	Norm. RMS yy	0.20 π.mm.mrad
	Norm. RMS

zz 0.12 π.M eV.deg

The optics in the MEBT is such that the quadrupoles gradients are listed in

Table 3

 3

and the rebunchers E 0 TL in Table

4

. The resulting MEBT configuration

Table 3 :

 3 Required gradient for the quadrupoles in a regular operation mode

		QP1 QP2 QP3 QP4 QP5 QP6 QP7 QP8
	gradient (T/m)	9.4	-9.7	7.1	-4.3	1.7	-3.3	6.4	-6.1

Table 4 :

 4 Required voltage for each rebuncher in a regular operation mode

	RB1 RB2 RB3
	Voltage (kV) 74.7 32.3 90.7

Table 5 :

 5 Pressures and gas compositions in the rebuncher at three different moments (before/during/after RF conditioning).

		100h pumping At the end of the RF conditioning After conditioning
	Pressure	6.5 • 10 -8	1.3 • 10 -7	2.1 • 10 -8
	H 2 (%)	20 %	75 %	30 %
	H 2 O (%)	80 %	25 %	70 %