Barbara Schormair

Jens Plag

Maria Kaffe

Nadine Gross

Darina Czamara

Walter Samtleben

Peter Lichtner

Andreas Ströhle

Ioannis Stefanidis

Andreas Vainas

Nadine Groß

Efthimios Dardiotis

George K Sakkas

Christian Gieger

Bertram Müller-Myhsok

Thomas Meitinger

Uwe Heemann

Georgios M Hadjigeorgiou

Konrad Oexle

MD Juliane Winkelmann
email: winkelmann@lrz.tu-muenchen.de

MEIS1 and BTBD9 -genetic association with RLS in end-stage renal disease

Keywords: restless legs syndrome, end-stage renal disease, genetic association studies

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Introduction

Restless legs syndrome (RLS) in end-stage renal disease (ESRD) is a common form of symptomatic RLS and is associated with increased mortality and reduced quality of life [START_REF] Winkelman | Restless legs syndrome in end-stage renal disease[END_REF].

The prevalence of RLS ranges between 18.4% [START_REF] Merlino | Sleep disorders in patients with end-stage renal disease undergoing dialysis therapy[END_REF] and 45.8% [START_REF] Siddiqui | Risk factors for restless legs syndrome in dialysis patients[END_REF] in ESRD patients of European descent. Both RLS in ESRD and idiopathic RLS (iRLS) are diagnosed based on the clinical presentation of four main symptoms. These are specified in the essential diagnostic criteria defined by the International RLS Study Group (IRLSSG) [START_REF] Allen | Restless legs syndrome: diagnostic criteria, special considerations, and epidemiology. A report from the restless legs syndrome diagnosis and epidemiology workshop at the National Institutes of Health[END_REF]: [START_REF] Winkelman | Restless legs syndrome in end-stage renal disease[END_REF] an urge to move the legs, usually accompanied by unpleasant sensations, (2) precipitation of symptoms by rest and inactivity, (3) symptom relief by movement, and (4) worsening appearance in the evening or at night. An abnormal increase in periodic limb movements in sleep (PLMS) further supports the diagnosis and is present in 80 to 89% of RLS patients [START_REF] Montplaisir | Clinical, polysomnographic, and genetic characteristics of restless legs syndrome: a study of 133 patients diagnosed with new standard criteria[END_REF].

For iRLS, a relatively strong genetic contribution has been demonstrated [START_REF] Winkelmann | Genetics of restless legs syndrome (RLS): State-of-the-art and future directions[END_REF]. Genome-wide association studies (GWAS) in iRLS cases from Germany and Iceland identified variants at four loci encompassing the genes MEIS1, BTBD9, MAP2K5/SKOR1, and PTPRD [START_REF] Winkelmann | Genome-wide association study of restless legs syndrome identifies common variants in three genomic regions[END_REF][START_REF] Schormair | PTPRD (protein tyrosine phosphatase receptor type delta) is associated with restless legs syndrome[END_REF][START_REF] Stefansson | A genetic risk factor for periodic limb movements in sleep[END_REF].

Replication studies in iRLS cases from the United States (US) and Europe (Czech Republic, Austria, and Finland) confirmed the association of MEIS1 and BTBD9. The MAP2K5/SKOR1 locus was replicated in the European cases but showed only a trend for association in the US cases [START_REF] Vilarino-Guell | A genetic risk factor for periodic limb movements in sleep[END_REF][START_REF] Kemlink | Replication of Restless Legs Syndrome Loci in Three European Populations[END_REF]. With regard to end-stage renal disease, association studies have identified more than 20 variants contributing to disease susceptibility [START_REF] O'seaghdha | Genetics of chronic kidney disease[END_REF]. There is no overlap between these variants and the known RLS-associated variants.

The pathophysiology of RLS in ESRD remains to be elucidated. Comparing clinical and biochemical parameters, such as duration and frequency of dialysis treatment or various components of phosphate and iron metabolism, in RLS-positive and RLS-negative ESRD patients yielded ambiguous results [START_REF] Siddiqui | Risk factors for restless legs syndrome in dialysis patients[END_REF][START_REF] Collado-Seidel | Clinical and biochemical findings in uremic patients with and without restless legs syndrome[END_REF][START_REF] Gigli | Restless legs syndrome in end-stage renal disease[END_REF][START_REF] Huiqi | Restless legs syndrome (RLS) in uremic patients is related to the frequency of hemodialysis sessions[END_REF][START_REF] Takaki | Clinical and psychological aspects of restless legs syndrome in uremic patients on hemodialysis[END_REF][START_REF] Araujo | Restless legs syndrome in end-stage renal disease: Clinical characteristics and associated comorbidities[END_REF][START_REF] Manna | Restless legs syndrome enhances cardiovascular risk and mortality in patients with end-stage kidney disease undergoing longterm haemodialysis treatment[END_REF].

If and to what extent, the susceptibility to RLS in ESRD is influenced by a genetic predisposition is not known. Since RLS in ESRD and iRLS are characterized by the same key symptoms, they possibly share genetic risk factors. Accordingly, we investigated the iRLSassociated variants [START_REF] Winkelmann | Genome-wide association study of restless legs syndrome identifies common variants in three genomic regions[END_REF][START_REF] Schormair | PTPRD (protein tyrosine phosphatase receptor type delta) is associated with restless legs syndrome[END_REF][START_REF] Stefansson | A genetic risk factor for periodic limb movements in sleep[END_REF] in a case-control association study of ESRD patients from Germany and Greece.

Subjects and Methods

Study populations and recruitment procedures

Germany (GER): ESRD patients on maintenance hemodialysis were recruited between January 2005 and August 2008 in 16 dialysis centres in Munich and the surrounding region.

ESRD patients who agreed to participate in the study answered a self-administered diagnostic questionnaire incorporating the four essential IRLSSG criteria [START_REF] Allen | Restless legs syndrome: diagnostic criteria, special considerations, and epidemiology. A report from the restless legs syndrome diagnosis and epidemiology workshop at the National Institutes of Health[END_REF][START_REF] Berger | Sex and the risk of restless legs syndrome in the general population[END_REF]. All patients who answered at least one question affirmatively were then examined by a clinician specializing in RLS (JP or NG) by face-to-face interview and classified as "RLS-positive" or "RLSuncertain" (if less than four diagnostic criteria were fulfilled) according to this interview.

Patients who had answered all diagnostic questions negatively in the initial questionnaire were classified as "RLS-negative" without a personal interview. Greece (GR): ESRD patients on maintenance hemodialysis were recruited between January 2007 and December 2009 in seven dialysis centres in Central and Northern Greece. In brief, patients were interviewed (first screening), using the four IRLSSG criteria, by a nephrologist (AV) or neurologist (ED) trained by RLS experts (GMH and IS) [START_REF] Allen | Restless legs syndrome: diagnostic criteria, special considerations, and epidemiology. A report from the restless legs syndrome diagnosis and epidemiology workshop at the National Institutes of Health[END_REF][START_REF] Hadjigeorgiou | Low RLS prevalence and awareness in central Greece: an epidemiological survey[END_REF]. The diagnosis of RLS was confirmed in a subsequent personal interview conducted by an expert neurologist (GMH or IS).

In both populations, family history of RLS (defined as the proband's report of at least one relative of up to the third degree affected by RLS) was recorded for RLS-positive patients.

Data on duration of dependence on dialysis and on weekly rate of dialysis was collected for all ESRD patients. In the German sample, age of onset of RLS was also recorded.

The study was approved by the respective local ethics committees in Germany and Greece.

All patients were informed about the aims and protocol of the study. Written informed consent was obtained from all patients.

SNP selection, genotyping, and quality control

We genotyped ten iRLS-associated SNPs (single nucleotide polymorphisms) across the four genomic regions [START_REF] Winkelmann | Genome-wide association study of restless legs syndrome identifies common variants in three genomic regions[END_REF][START_REF] Schormair | PTPRD (protein tyrosine phosphatase receptor type delta) is associated with restless legs syndrome[END_REF][START_REF] Stefansson | A genetic risk factor for periodic limb movements in sleep[END_REF]. Genotyping was performed according to the iPLEX Gold protocol using matrix-assisted laser desorption/ionization time-of-flight (MALDI-TOF) mass spectrometry (Sequenom). Assays were designed using AssayDesign 3.1.2.2 with the default parameters for the iPLEX Gold chemistry. Cleaned extension products were analysed by mass spectrometer (Bruker Daltronik) and peaks were identified using SpectroTYPER RT 3.4.

Quality control criteria leading to exclusion of a SNP from further analysis were a call rate < 95%, a minor allele frequency (MAF) < 5%, and a P value (P) < 0.001 for deviation from Hardy-Weinberg-Equilibrium (HWE) in controls. Furthermore, DNA samples with a call rate < 95% over all SNPs were excluded from the analysis.

Statistical analysis

Statistical analysis was performed using PLINK v1.07 [START_REF] Purcell | PLINK: a tool set for whole-genome association and population-based linkage analyses[END_REF] for the SNP association tests and R 2.10.1 (www.r-project.org) for all other analyses. Statistical significance was defined at the 95% level (P < 0.05). Association of disease status and SNP allele count was tested using logistic regression with age and gender as covariates. For the combined analysis of both populations, the sample was stratified based on the country of origin and a Cochran-Mantel-Haenszel test was applied. To correct for multiple testing, empirical P values were generated by the max(T) permutation procedure implemented in PLINK v1.07 (n = 10,000).

Dialysis parameters and mean age were compared between cases and controls by means of a two-sided Student's t-test. A χ 2 -test was utilized to compare gender ratios. Study power was calculated using the CATs power calculator [START_REF] Skol | Joint analysis is more efficient than replication-based analysis for two-stage genome-wide association studies[END_REF], with settings of an additive genetic model, disease prevalence of 0.2, and risk allele frequencies and odds ratios (ORs) as estimated in previous studies [START_REF] Winkelmann | Genome-wide association study of restless legs syndrome identifies common variants in three genomic regions[END_REF][START_REF] Schormair | PTPRD (protein tyrosine phosphatase receptor type delta) is associated with restless legs syndrome[END_REF][START_REF] Kemlink | Replication of Restless Legs Syndrome Loci in Three European Populations[END_REF].

Results

Study populations

In the German study, 737 of 1,617 ESRD patients agreed to participate (45.6%). Many ESRD patients had comorbidities such as dementia or depression, and, therefore, were not able to participate, leading to the relatively low response rate. According to the diagnostic questionnaire, 253 ESRD patients were considered "potential RLS" in ESRD cases. Of these, 53 were subsequently excluded from the study (uncertain diagnosis: n = 44; non-European descent: n = 6; incomplete data: n = 3). The final study population included 200 RLS-positive patients (prevalence of RLS in ESRD = 31.1%). Positive family history of RLS was reported by 38 of these patients (19%), 133 (66.5%) had a negative family history, and for 29 (14.5%) there was no information available. Controls were selected from 484 RLS-negative ESRD patients. Of these, 41 were excluded from the study (non-European descent: n = 13, incomplete data: n = 28), yielding a final control sample of 443 RLS-negative ESRD patients.

Cases and controls did not differ significantly with regard to age, gender ratio, and the recorded dialysis parameters (Table 1 1).

Comparison of demographic data and dialysis parameters between the German and the Greek sample showed significant differences in the weekly rate of dialysis and the mean duration of dependence on dialysis whereas mean age and gender ratios were similar (Table 1). Loci are ordered according to the power of the study to detect the effects, based on allele frequencies and ORs found in previous studies [START_REF] Winkelmann | Genome-wide association study of restless legs syndrome identifies common variants in three genomic regions[END_REF][START_REF] Schormair | PTPRD (protein tyrosine phosphatase receptor type delta) is associated with restless legs syndrome[END_REF][START_REF] Kemlink | Replication of Restless Legs Syndrome Loci in Three European Populations[END_REF]. MAF, minor allele frequency; Ca, cases; Co, controls; P nom , nominal P value from logistic regression analysis with age and gender as covariates. For the combined analysis, country of origin was added as an additional covariate. P corrected , P value corrected for multiple testing, empirical P value obtained from permutation. ORs, odds ratios, were obtained from logistic regression analysis and refer to the risk allele. Significant associations are highlighted in bold print.

A subanalysis in the German sample with cases stratified according to their family history revealed a trend for differences in the size of the contribution of the associated loci to familial versus sporadic RLS in ESRD. Analysis of cases with a positive family history (n = 38) only, revealed a significant association both to MEIS1 and BTBD9 (rs12469063, P nom = 0.003, P corrected = 0.014; rs2300478, P nom = 0.005, P corrected = 0.023, and rs3923809, P nom = 0.006, P corrected = 0.03). Using only cases with a negative family history (n = 133), the two loci showed nominally significant P values (rs12469063, P nom = 0.038, P corrected = 0.224; rs2300478, P nom = 0.055, P corrected = 0.306, and rs3923809, P nom = 0.025, P corrected = 0.147).

Odds ratios differed for both groups but the 95% confidence intervals (CI) overlapped:

Positive family history: MEIS1: 2.1 (1.29-3.39); BTBD9: 2.39 (1.28-1.87); negative family history: MEIS1: 1.38 (1.02-1.87); BTBD9: 1.42 (1.05-1.92). The Greek sample did not contain enough cases with a positive family history (n = 7) to conduct a similar analysis.

Discussion

It has long been known that idiopathic and symptomatic RLS share the same key symptoms and that iRLS susceptibility is subject to genetic influences. Here, we show that genetic predisposition also plays a role in RLS in ESRD and that at least some of the underlying risk variants are common to both iRLS and RLS in ESRD.

In the German sample, sequence variants in MEIS1 and BTBD9 were clearly identified as genetic susceptibility factors for RLS in ESRD patients (P corr ≤ 0.03). The effect size of these variants is within the same range as observed in iRLS studies of comparable sample size (ORs RLS in ESRD = 1.47 to 1.55, 95% CIs = 1.19-2.04; ORs iRLS = 1.43 to 1.59, 95% CIs = 1.12-2.2) [START_REF] Vilarino-Guell | A genetic risk factor for periodic limb movements in sleep[END_REF][START_REF] Kemlink | Replication of Restless Legs Syndrome Loci in Three European Populations[END_REF]. This was confirmed in an analysis using population-based controls as the control group yielding similar odds ratios and confidence intervals for the associated genes MEIS1 and BTBD9 (data not shown). However, the present data did not indicate a contribution of variants in MAP2K5/SKOR1 and PTPRD to RLS in ESRD in the German sample. The absence of significant association to these loci may be due to the relatively small sample sizes in our study, i.e. to the limited power.

Concerning the Greek sample, we could observe a trend for association to BTBD9 and MAP2K5/SKOR1. Several differences between both samples could account for the divergences between the German and the Greek results. i) Sample sizes differed. ii) Greek ESRD patients were on dialysis for a significantly shorter time than the German patients and at a lower weekly rate (Table 1). In several previous studies, the frequency of RLS in ESRD increased with the duration of dependence on dialysis [START_REF] Siddiqui | Risk factors for restless legs syndrome in dialysis patients[END_REF][START_REF] Collado-Seidel | Clinical and biochemical findings in uremic patients with and without restless legs syndrome[END_REF][START_REF] Gigli | Restless legs syndrome in end-stage renal disease[END_REF][START_REF] Huiqi | Restless legs syndrome (RLS) in uremic patients is related to the frequency of hemodialysis sessions[END_REF][START_REF] Takaki | Clinical and psychological aspects of restless legs syndrome in uremic patients on hemodialysis[END_REF][START_REF] Araujo | Restless legs syndrome in end-stage renal disease: Clinical characteristics and associated comorbidities[END_REF][START_REF] Manna | Restless legs syndrome enhances cardiovascular risk and mortality in patients with end-stage kidney disease undergoing longterm haemodialysis treatment[END_REF]. It is possible, therefore, that some of the currently RLS-negative Greek ESRD patients will develop RLS in the future. Such false-negative controls would distort the association results. iii) The samples differed markedly in the percentage of RLS cases with a positive family history (19% in GER versus 5% in GR). The subanalysis based on family history of RLS in the German sample suggested a stronger contribution of the genetic risk factors in cases with a positive family history. This trend has also been observed in iRLS populations [START_REF] Vilarino-Guell | A genetic risk factor for periodic limb movements in sleep[END_REF][START_REF] Kemlink | Replication of Restless Legs Syndrome Loci in Three European Populations[END_REF]. The lower percentage of familial RLS could thus explain the weakened association signal for BTBD9 in the Greek sample.

However, it has to be considered that the information on family history was obtained only by the proband's report, and was not confirmed in a direct personal interview of the relatives. Therefore, both false-negative and false-positive classifications may have occurred. iv) For MEIS1, we see a sizeable difference in allele frequency between Greek and German RLSpositive ESRD patients (Table 2) indicating that in the Greek population either MEIS1 is not an RLS risk factor or the causal variant in MEIS1 may not be in linkage disequilibrium with the variants under study.

Despite the differences in sample characteristics and individual association results, BTBD9

shows a consistent trend for association in the separate as well as the combined samples.

Previous analysis in iRLS has linked BTBD9 to the presence of PLMS, the motor component of RLS [START_REF] Stefansson | A genetic risk factor for periodic limb movements in sleep[END_REF]. Therefore, its association to RLS in ESRD is remarkable because motor symptoms seem to be more prominent in RLS in ESRD when compared to iRLS [START_REF] Wetter | Polysomnographic sleep measures in patients with uremic and idiopathic restless legs syndrome[END_REF][START_REF] Enomoto | Clinical characteristics of restless legs syndrome in end-stage renal failure and idiopathic RLS patients[END_REF].

The prevalence of RLS in ESRD observed in our study (31.1% in GER, 26.4% in GR) is in keeping with previous investigations [START_REF] Merlino | Sleep disorders in patients with end-stage renal disease undergoing dialysis therapy[END_REF][START_REF] Gigli | Restless legs syndrome in end-stage renal disease[END_REF][START_REF] Huiqi | Restless legs syndrome (RLS) in uremic patients is related to the frequency of hemodialysis sessions[END_REF][START_REF] Takaki | Clinical and psychological aspects of restless legs syndrome in uremic patients on hemodialysis[END_REF][START_REF] Araujo | Restless legs syndrome in end-stage renal disease: Clinical characteristics and associated comorbidities[END_REF][START_REF] Manna | Restless legs syndrome enhances cardiovascular risk and mortality in patients with end-stage kidney disease undergoing longterm haemodialysis treatment[END_REF] and higher than that of RLS in the general population at the same age (10% in GER, 3.9% in GR) [START_REF] Rothdach | Prevalence and risk factors of RLS in an elderly population: the MEMO study. Memory and Morbidity in Augsburg Elderly[END_REF][START_REF] Phillips | Epidemiology of restless legs symptoms in adults[END_REF][START_REF] Hadjigeorgiou | Low RLS prevalence and awareness in central Greece: an epidemiological survey[END_REF]. The prevalence of iRLS is substantially lower in Greece, while the prevalence of ESRD is quite similar with prevalence rates of 0.11% and 0.10% in Germany and Greece, respectively [START_REF]U S Renal Data System, USRDS 2009 Annual Data Report: Atlas of Chronic Kidney Disease and End-Stage Renal Disease in the United States[END_REF].

The effect sizes of MEIS1 and BTBD9 in our study are of the same magnitude as in iRLS and, thus, do not account for the increased frequency of RLS in ESRD patients. It is likely that there are as yet undiscovered genetic, non-genetic, or non-shared environmental factors influencing the RLS phenotype. Our finding that only 5% (GR) to 19% (GER) of patients with RLS in ESRD reported a positive family history is in line with previous observations showing a lower frequency of familial RLS (12%) [START_REF] Winkelmann | Clinical characteristics and frequency of the hereditary restless legs syndrome in a population of 300 patients[END_REF] in cases with RLS in ESRD compared to iRLS cases (30 -92%) [START_REF] Winkelmann | Genetics of restless legs syndrome (RLS): State-of-the-art and future directions[END_REF]. The reduced familial clustering of RLS in ESRD patients could be interpreted as an indicator of a stronger influence of non-genetic factors in the aetiology of RLS in ESRD as compared to iRLS. Moreover, studies have shown that RLS symptoms are reduced after renal transplantation [START_REF] Molnar | Restless Legs Syndrome in patients after renal transplantation[END_REF], suggesting that non-genetic factors such as reduced renal function and incomplete clearance of uremic toxins during dialysis have a major influence on the development of RLS in ESRD.

In conclusion, we were able to show a genetic contribution to susceptibility for RLS in ESRD.

BTBD9 is significantly associated with RLS in ESRD. Our results suggest that the genetic influence might be stronger in the subgroup of RLS in ESRD patients with positive family history. The study was limited to a subset of genes previously identified to be associated with iRLS. A genome-wide association study using RLS-positive (cases) and RLS-negative (controls) ESRD patients might disclose further genetic risk variants for RLS in ESRD. All or some of these could -albeit not necessarily -also play a role in iRLS.

  ). In the Greek study, 572 of 613 ESRD patients agreed to participate (93.3%). According to the first screening, 162 ESRD patients were considered potential RLS in ESRD cases. Of these, 21 were subsequently excluded from the study (uncertain diagnosis: n = 15; non-European descent: n = 1; incomplete data: n = 5) resulting in a case sample of 141 RLS-positive patients (prevalence of RLS in ESRD = 26.4%). Positive family history of RLS was reported by seven patients (4.9%), 129 (91.5%) had a negative family history, and for five (3.6%) no information was available. Controls were selected from 410 RLS-negative ESRD patients. After excluding 17 patients due to incomplete data, 393 RLS-negative ESRD patients Schormair, Plag et al. 8 remained. Cases and controls had a significantly different mean age, but were similar in gender ratio and dialysis parameters (Table

Table 1 :

 1 Demographic and dialysis data of final study population.

	GER

Table 2 :

 2 Association results.

	GER	GR	Combined analysis

Acknowledgement

We are grateful to all patients who participated in this study. We also thank the Kuratorium

Dialyse und Nierentransplantation e.V. (KfH) dialysis centres KfH Nierenzentrum Ebersberg (Dr. med. Kathrin Stephanie Lange), KfH Nierenzentrum Freising (Dr. med. Monika Wildgruber), KfH Nierenzentrum Germering (Dr. med. Hermann Hacker), KfH Nierenzentrum Ingolstadt (Dr. med. Friedrich Josef Lazarus), KfH Nierenzentrum Miesbach (Dr. med. Daniela Anne Soreth-Rieke), KfH Nierenzentrum München Burggrafenstrasse (Dr. med. Gabriele Schätzle), KfH Nierenzentrum München Elsenheimer Strasse (PD Dr. med. Stephan Lederer), KfH Nierenzentrum München Isoldenstrasse (Prof. Dr. med. Johannes Mann), KfH Nierenzentrum München Seybothstrasse (Prof. Dr. med. Jürgen Scherberich), KfH Nierenzentrum Neuried (Prof. Dr. med. Walter Samtleben), KfH Nierenzentrum Oberschleissheim (Dr. med. Norbert Bockreiss), KfH Nierenzentrum Pfaffenhofen (Dr. med. Axel Witte), KfH Nierenzentrum Rosenheim (Dr. med. Andreas Thiele), KfH Nierenzentrum Straubing (Prof. Dr. med. Marianne Haag-Weber), KfH Nierenzentrum Traunstein (Dr. med.

Martin Gottsmann), and KfH Nierenzentrum Unterhaching (Dr. med. Klaus Hällfritzsch) for their help in conducting the study. We also thank the participating patients and dialysis centres in Greece.

Association analysis

The ten selected SNPs passed quality control and were subjected to statistical analysis. A total of 43 individuals (22 in GER sample, 21 in GR sample) were excluded due to low genotyping quality. Analysis of the German sample (195 cases and 426 controls) revealed significant association to RLS in ESRD for variants in MEIS1 and BTBD9 (Table 2). Within MEIS1, two of three SNPs were significantly associated after correction for multiple testing: rs12469063 (P nom = 0.002, P corrected = 0.013, OR = 1.52) and rs2300478 (P nom = 0.004, P corrected = 0.026, OR = 1.47). In BTBD9, rs3923809 was associated (P nom = 0.001, P corrected = 0.009, OR = 1.56).

For MAP2K5/SKOR1 and PTPRD the nominal P values ranged from 0.17 to 0.69 (Table 2). In contrast, the Greek sample (138 cases, 375 controls) showed only nominal significance for a single SNP in MAP2K5/SKOR1 (rs3784709, P nom = 0.021) and a trend of association for two further SNPs in MAP2K5/SKOR1 and for rs3923809 in BTBD9 (Table 2). In the combined analysis of all samples, stratified according to the country of origin, rs3923809 in BTBD9 and four SNPs in MAP2K5/SKOR1 showed nominal significance but only rs3923809 remained significant after correction for multiple testing (Table 2).

The statistical power of the German sample was higher than that of the Greek sample, owing to the larger sample size. For MEIS1, power in both samples was ≥ 90%, for BTBD9 it was 85% in the German and 74% in the Greek sample. For the other loci, power was < 80% in both populations (Table 2).

Competing financial interests:

The authors declare no competing financial interests.

Funding: This study was supported in part (for GMH and ET) from University of Thessaly, Research Committee (Code: 2845).

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N R Badnell

A Foster

‡ D C Griffin

D Kilbane

M O'mullane

H P Summers

6 4d q -4p 6 4d (q-1) 4f + 4p 5 4d (q+1) transition arrays for q = 1 -10

Keywords:

come

Dielectronic recombination of heavy species: the tin 4p64dq 4p64d(q 1)4f + 4p54d(q + 1) transition arrays for q = 110 . Dielectronic recombination of heavy species: the tin 4p64dq 4p64d(q 1)4f + 4p54d(q + 1) transition arrays for q = 110. Journal of Physics B: Atomic, Molecular and Optical Physics, 2011, 19 (13), pp.135201-3835. 10.1088/0953-4075/44/13/135201. hal-00632297

Introduction

Tin plasmas are a keen interest of study as an extreme ultra-violet (EUV) light source for next generation microlithography [START_REF] Bakshi | EUV Sources for Lithography[END_REF]. Usable wavelengths are rather restricted due to mirror absorption etc. Suitable emission around 13.5 nm is particularly intense from Sn 7+ through Sn 12+ ions [2]. It is dominated by the 4p 6 4d q -4p 6 4d (q-1) 4f+4p 5 4d (q+1) unresolved transition array (UTA). Such an arrangement of configurations is subject to strong configuration-mixing which leads to a spectacular quenching of the unmixed spectra [3,4]. Modelling such emission requires a knowledge of the tin ionization balance. The dominant recombination mechanism is dielectronic. Its accurate description is somewhat demanding due to the complexity of the configurations involved -the open d-subshell targets. The only detailed calculation we can find is on Sn 10+ [5] which made use of the Flexible Atomic Code (FAC) [6].

Heavy species are also of particular interest to magnetic fusion and especially ITER [START_REF]ITER Physics Basis[END_REF]. A tin-coated probe has been used on the Mega Amp Spherical Tokomak (MAST) to study heavy species markers for erosion transport diagnostics [8]. The main heavy species for ITER will be Tungsten. It is in this context that Ballance et al [9] studied the dielectronic recombination (DR) of W 35+ -which is isoelectronic with Sn 11+ . They carried-out calculations using three different methods viz. Dirac R-matrix close-coupling, Breit-Pauli perturbation theory (via the autostructure code) and configuration-average (CA). The configuration-average total dielectronic recombination rate coefficients were only 10-15% larger than those obtained from perturbation theory while the R-matrix ones were ∼ 20% smaller than perturbation theory's. Configuration-mixing was omitted from the level-resolved calculations as it was too demanding computationally. (The configuration-average method 'omits' configuration-mixing fundamentally.) It is of interest to see how well the configuration-average method fares once configuration-mixing is taken into account by the detailed (perturbation theory) calculations. The configuration-average description of dielectronic recombination is very efficient and can be applied easily to all ionization stages of tin and tungsten and can take account of all significant recombination pathways configuration-resolved. It is also an improvement over the present modelling usage of average-atom rate coefficients -these were nearly a factor of two larger than configuration-average for W 35+ at typical magnetic fusion plasma temperatures [9].

Recent developments to the autostructure code have significantly reduced its memory requirement for complex heavy species [10] and it is now possible to treat dielectronic recombination via the 4p 6 4d q -4p 6 4d (q-1) 4f+4p 5 4d (q+1) transition arrays for q = 1 -10 in a full configuration-mixed level-resolved Breit-Pauli calculation using kappa-averaged relativistic wavefunctions. This opens-up its application more widely to technical, fusion [11] and astrophysical plasmas [12] involving heavy specieswhich we take to be elements of the periodic table which lie beyond Zinc.

It is desirable also to consider an additional approach which falls between the full level-resolved and configuration-averaged ones -the Burgess-Bethe [13]. This approach is the one used by the Atomic Data and Analysis Structure (ADAS) [START_REF] Summers | The ADAS USer Manual[END_REF] for its baseline description of dielectronic recombination. The baseline approach is the one used for the collisional-radiative modelling of heavy species for magnetic fusion [8]. The Burgess-Bethe method [13] treats dielectronic recombination in an uncoupled representation (between the target and colliding/Rydberg electron) and thus is able to take account of configuration-mixing in the N -electron target/core alone. Thus it is superior to the configuration-average method and much less demanding computationally than the full level-resolved perturbation approach of autostructure which describes (N + 1)-electron configuration-mixing in the dielectronic recombination process. The original implementation in the Burgess-Bethe General Program (BBGP) was the basis for the seminal Burgess General Formula [START_REF] Burgess | [END_REF]. It has undergone significant enhancement to treat 'alternative' pathways [16] which were not described back then.

The outline of the paper is as follows: in section 2 we describe briefly the theory behind the perturbative treatment of dielectronic recombination as it is implemented in the autostructure code, the Burgess-Bethe General Program and the configuration-average; in section 3 we compare results from these three different methods; in section 4 we recapitulate; and in section 5 we draw some conclusions.

Theory

The partial dielectronic recombination rate coefficient α z f i from an initial state i of an ion X +z into a resolved final state f of an ion X +z-1 is given by

α z f i (T ) = 4πa 2 0 I H k B T e 3/2 j ω j 2ω i e -Ec/(kBTe) × l A a j→i,Ecl A r j→f h A r j→h + m,l A a j→m,Ecl (1)
in the isolated resonance approximation [13]. Here ω j is the statistical weight of the (N + 1)-electron doubly-excited resonance state j, ω i is the statistical weight of the N -electron target state (so z = Z -N ) and the autoionization (A a ) and radiative (A r ) rates are in inverse seconds. E c is the energy of the continuum electron (with orbital angular momentum l) which is fixed by the position of the resonances and I H is the ionization potential energy of the hydrogen atom (both in the same units of energy), k B is the Boltzman constant, T e the electron temperature and (4πa 2 0 ) 3/2 = 6.6011×10 -24 cm 3 .

The spectroscopic modelling of finite-density plasmas requires final-state resolution so as to take account of collisional redistribution of the final state. This is beyond the scope of the present paper and so all of the results that we present are for the total recombination rate coefficient i.e. summed-over all f that are stable against autoionization.

Equation ( 1) is applicable to all three methods that we consider. The difference lies in the interpretation of the state representation: level-resolved, uncoupled parentplus-electron or configuration.

Implementation

All three methods include only the electric dipole radiation operator in the DR process. autostructure describes arbitrary E k and M k multipoles but they are of negligible importance for total dielectronic recombination rate coefficients. All three methods include only the electrostatic operator to describe autoionization. autostructure describes both the two-body fine-and non-fine-structure operators of the Breit-Pauli Hamiltonian as well. They can be neglected here.

autostructure

The state labels are levels and are readily interpretable in the usual sense. Explicit expressions for the autoionization rate may be found in [17] and the radiative rate in [18].

Calculations were carried-out for Rydberg nl up to n = 100 and l = 7 so as to converge the total dielectronic recombination rate coefficients to ∼ 1%. We included the three target configurations of the title which are then coupled to Rydberg and continuum electrons. We also included all (N +1)-electron bound-orbital configurations that result from adding an n = 4 orbital to the target configurations. This describes outer electron radiation into the core. Radiation from n → n ≥ 5 is treated hydrogenically. This is the only additional approximation to the Breit-Pauli level-resolved isolated resonance approach.

We note that we form the N -and (N + 1)-electron interaction Hamiltonians explicitly all the way up to n = 100 on making use of the Correspondence Principle. This enables us to generate autoionization rates for channels which do not open-up until high-n (e.g. n = 40). These autoionizing transitions between fine-structure levels § strongly suppress recombination through the high-n states. Extrapolation of autoionization rates from low-n (e.g. n = 15) omits these channels and can lead to a substantial overestimate of the high temperature (total) DR rate coefficients. There is a converse effect. If the ground term splits into fine-structure levels then lowtemperature dielectronic recombination can be enhanced significantly by these high-n dielectronic captures. It is a highly-charged ion effect since the radiative stabilization is by the Rydberg electron to a large set of true bound states built upon excited parents [19].

We use the kappa-averaged relativistic orbital approximation [20] but with a selfconsistent configuration-average model potential [START_REF] Cowan | The Theory of Atomic Structure and Spectra[END_REF] rather than the Hartree-Fock Relativistic (HFR) [20]. The required occupation numbers are the average of the three target configurations with the appropriate self-interaction omitted for each orbital i.e. the potentials are slightly different for each orbital. The kappa-averaged approach inherently gives rise to non-orthogonal orbitals and so there is nothing to be gained by using a unique model potential. We neglect the small component of the Dirac spinor.

The advantages of this approach are that all possible pathways are described levelresolved with full configuration-mixing in both the N -and (N + 1)-electron bound orbital configurations. The disadvantage is that it is computationally demanding and so places restrictions on the number of N -electron configurations used to describe the problem. It is largely a memory limitation. Each Rydberg nl is treated separately and the code is already parallel by l. The wallclock time could be reduced further by parallelization in n. The trend of modern computers is towards massive parallelization but with relatively small memory per processor. Node memory is then fully used without full utilization of the available processors.

2.1.2.

BBGP The state labels are target/parent levels uncoupled to the Rydberg/continuum electron in a bundled-nl representation. Explicit expressions for the radiative and autoionization rates may be found in [16]. They are given in terms of an LS-coupling representation but are readily recoupled to LSJ. The use of the Bethe (dipole) approximation for the autoionization rates greatly simplifies equation [START_REF] Bakshi | EUV Sources for Lithography[END_REF]. It reduces the problem to an N -electron description for the radiative rates plus rapidly evaluated hydrogenic bound-free radial integrals. The inaccuracy of the Bethe approximation for low-l is overcome through the introduction of 'universal' correction factors. The correction factors and their use have been improved-upon over the years. The present BBGP implementation [16] also allows for alternate Auger break-up and parent radiative transition pathways that were not part of the original one by Burgess [13]. A further refinement is possible if (threshold) distorted wave partial collision strengths are available from some external source such as the new development in autostructure [10]. Non-dipole autoionization pathways can be § The autoionization interaction itself is still the Coulomb electrostatic. described as well then and the dipole rates can use explicit correction factors instead of the 'universal' ones. The present work is restricted to dipole transitions only.

Calculations were carried-out to convergence of the Rydberg nl (typically to n = 1000 and l = 15) for both ∆n p = 0 and ∆n p = 1 dipole promotions from the initial ground configuration. The bound-state wavefunctions, radiative rates and energies are determined in the HFR approximation [20] using a modified version of the RCN code developed by Cowan [START_REF] Cowan | The Theory of Atomic Structure and Spectra[END_REF].

The advantages of the BBGP approach are its ability to determine rapidly the relative contributions from all possible recombination pathways for arbitrary complex ions and its inclusion of N -electron configuration-mixing. Limitations of the BBGP approach are the restriction to dipole autoionizing transitions unless external distorted wave data is available and the neglect of (N + 1)-electron configuration-mixing.

CA

The state labels are configurations and so ω i and ω j are the total statistical weights of the N -and (N + 1)-electron configurations. The configuration-average approximation method referred to here is an a priori analytic average. This is not to be confused with the a posterior numerical average of term/level resolved (configuration-mixed) atomic data for subsequent 'CA' modelling. Explicit expressions for the configuration-average rates used here may be found in [START_REF] Griffin | [END_REF][START_REF] Pindzola | Atomic Processes in Electron-Ion and Ion-Ion Collisions[END_REF]. The program implementation DRACULA (dielectronic recombination average configuration using local approximation) has been extensively revised recently [9] to handle up to eight open subshells and for a more robust treatment of high-nl rates.

The present calculations were carried-out up to n = 1000 and l = 12. The boundstate wavefunctions and energies are determined in the Hartree-Fock Relativistic (HFR) approximation [20] using a slightly modified version of the latest RCN code (Mod36) developed by Cowan [START_REF] Cowan | The Theory of Atomic Structure and Spectra[END_REF].

The main advantage of the configuration-average approach is the ability to determine rapidly the relative contributions from all possible recombination pathways for arbitrary complex ions. Limitations of the approach and/or implementation: a configuration which straddles the ionization limit must be taken to be either bound or autoionizing, autoionization within a core configuration (e.g. 4d q nl → 4d q + e -) is omitted and configuration-mixing is not represented in any way.

Configuration-mixing

We demonstrate the strong mixing between the 4p 6 4d (q-1) 4f and 4p 5 4d (q+1) configurations. We choose to do so for Sn 10+ (q = 4) since it (along with Sn 11+ ) is the strongest emitter [2] around 13.5 nm. Similar effects occur for all q considered here. In figure 1 we show the total line strength emission as function of wavelength. We have convoluted it with a 0.01 nm FWHM Gaussian so as to facilitate comparison. We show the emission from the 4p 6 4d 3 4f and 4p 5 4d 5 configurations calculated separately and from the two mixed. The latter is clearly not a superposition of the former two. The spectral emission has changed dramatically. Such effects have been observed before -in Pr 21+ for example [4]. It is important to remember that the total line strength emission summed-over all wavelengths is unchanged since the only difference between the two sums is a pure unitary transformation. Higher-order energy moments would differ somewhat since the differing wavelengths then come into play.

We consider now the implication for dielectronic recombination as described by equation [START_REF] Bakshi | EUV Sources for Lithography[END_REF]. Autoionization rates can be expected to be changed markedly by configuration-mixing in a similar fashion to the radiative rates. This is exemplified by the Bethe approximation as we have noted already. If all autoionizing levels satisfy either A r << A a or A r >> A a then to a first approximation all Auger yields or fluorescence yields are unity. The total dielectronic recombination rate coefficient is unchanged then by the unitary configuration-mixing transformation save for the exponential factor. The exponential is unity at high temperatures. At low temperatures any near threshold autoionizing levels will have their relative position changed dramatically by configuration-mixing within the core. The energy factor then changes exponentially. The change in wavelength in figure 1 is due purely to changes in position of the upper level. The position of the lower levels is unaffected since we consider only electric dipole transitions.

The effect of configuration-mixing on the total dielectronic recombination rate coefficient then depends on the degree to which the above conditions are violated. Individual partial final-state resolved dielectronic recombination rate coefficients can be expected to change dramatically along the lines illustrated in figure 1. These are necessary for spectral diagnostics and finite-density collisional-radiative modelling.

Results

We establish first the dominant contributions to the tin dielectronic recombination rate coefficients using the configuration-average method and then compare and contrast It depends as well on (the sum over) i, f, h and m -see the detailed study by Cowan and Griffin [24] for further details. them with the results of our level-resolved calculations. We study Sn 11+ first since this is isoelectronic with W 35+ and so we can compare and contrast with the findings of [9]. We next look at the lower charge end and the case of Sn 7+ . We consider then the remaining ions of interest as an EUV light source: Sn 8+ -Sn 10+ , Sn 12+ and Sn 13+ . The lowest-charge 4d q (q = 8 -10) Sn ions are considered also since they are of interest in establishing the ionization balance for magnetic fusion diagnostics: the closed-subshell case of Sn 4+ is studied in detail.

Sn 11+

In figure 2 we show the configuration-average results for the total contribution from ∆n = 0 and ∆n = 1 promotions from the ground configuration (4p 6 4d 3 ). ¶ We show also the contribution from the dipole promotions alone viz. 4p -4d plus 4d -4f. We see that the ∆n = 1 contribution is negligible from a plasma modelling perspective -it is at most 10% at the point where the ∆n = 0 contribution has a minimum. We see also that the dipole promotions dominate the ∆n = 0 contribution. Both of these results are in contrast to the behaviour seen for W 35+ [9]. The difference in charge-state explains both trends and they are also seen in L-and M -shell DR: a higher charge-state leads to relatively larger radiative rates compared to autoionization as well as to core-excited states supporting more true bound states. In the case of W 35+ the non-dipole promotion 4p -4f contributed about a third of the ∆n = 0 contribution because the core radiative stabilization 4f -4d led to true bound states of the form 4p 5 4d 4 nl (for n = 8 -10 depending on l). Fewer such states contribute at lower charge. The net result can be seen in figure 3 where we show the contributions from 4l -4l . We see that the non-dipole 4p -4f promotion makes a non-negligible contribution only where there is a minimum in the dipole contribution. These relative contributions should be contrasted with those of [9] figure 3. Similar behaviour is seen for the ∆n = 1 promotions shown in figure 4. The dipole promotions 4p -5s and 4d -5p dominate and this is in contrast to that seen for W 35+ -see [9] figure 5.

We turn now to comparing level-resolved results from autostructure with those from the configuration-average. We calculated the 4p-4d and 4d-4f separately first so as to omit the strong configuration-mixing. The question arises then as to which initial levels to average-over. In [9] it is argued that it is unlikely that just the ground level is populated in a collisional plasma. Thus they averaged-over all initial levels of the ground configuration and stated "This allowed for a direct comparison with the CADW results and also should be more appropriate for modelling." We agree that excited levels may well have significant populations but it remains to be determined whether they are in any way statistical -except perhaps for the case of the fine-structure levels within a term. It is hard also to say which is more appropriate for comparing with configuration-average. The configuration-average method cannot describe re-autoionization into alternative continua of the ground configurationthere is only one. In the level-resolved case dielectronic capture from the ground level can re-autoionize into progressively excited continua only from increasingly large n. But dielectronic capture from an excited level can always re-autoionize to the continuum of the ground level. The net result is that dielectronic recombination from increasingly excited initial levels is increasingly smaller than that from the ground ¶ The distinction between ∆n = 0 and ∆n = 1 on energy grounds starts to become blurred at these charge states. level. Which is a more meaningful comparison with the configuration-average is not clear.

In figure 5 we present our 'unmixed' 4p -4d and 4d -4f contributions from autostructure averaged-over just the statistical weight of the ground level as well as for the whole configuration -the former being 50% larger than the latter at the high temperature peak. The configuration-average results for 4p -4d and 4d -4f are 30% and 60% larger than the autostructure ones averaged over the initial ground level. They are just over a factor of two larger than the autostructure results averaged-over all levels of the ground configuration. (The large low temperature contributions have a modest influence of ∼10% at the high temperature peak. Discounting them slightly increases the CA 'overestimate' since its low temperature contribution is smaller.) The closer agreement of the configuration-average results with those for averaging-over only the ground level is presumably since this 'minimizes' the suppression by autoionization into excited states. From a modelling perspective we would need to examine the collisional (-radiative) excitation rates so as to determine which of the initial levels were populated significantly. With initial-level-resolved dielectronic recombination data to hand then we can tailor it to the local plasma conditions.

We consider next the inclusion of both dipole promotions in the same autostructure calculation. We find a net increase in their total rate coefficient of only 10% at the high temperature peak compared to summing the two treated separately. This is a combination of the effects of configuration-mixing and the additional autoionization channels between the two excited configurations. Unitarity has a significant role in suppressing differences due to mixing in total rate coefficients. This is less so for the case of final-state resolved partial rate coefficients.

In figure 6 we compare our configuration-mixed level-resolved autostructure results with the total rate coefficients from the BBGF and CA methods. The first two sets of results are averaged-over the initial ground level and the latter two include all significant promotions. The three sets of results have very different low temperature contributions. This is not surprising but it slightly complicates comparisons made at the high temperature peak. We can estimate the contribution for the tin ions considered in this paper as follows based-on the relevant scalings with temperature illustrated by equation ( 1): we take the rate coefficient at ∼10 eV and reduce it by a factor of 10 and then take this to be the estimate of the low temperature contribution to the high temperature peak at ∼50 eV. About 8% of the autostructure total arises this way while it is negligible for the BBGP and CA results. This 'additional' autostructure contribution roughly cancels the contribution from the 'additional' promotions included in the BBGP and CA results over and above the former. Then the CA results are about 45% larger and the BBGP 20% smaller than the autostructure results at the high temperature peak. If we were to average over all levels of the ground configuration then the CA results would now lie significantly higher than the other two sets -recall figure 5. Results from the Burgess General Formula (GF) are also shown. These were determined using the 4p -4d and 4d -4f oscillator strengths from the configuration-mixed level-resolved autostructure target. The peak temperature result is similar to the CA. We note that all results show a large variation at low temperatures. The relevant temperatures of interest for technical plasmas are 30 -50 eV while for magnetic fusion plasmas they are somewhat higher at 100 -200 eV.

In figure 7 we show the total collisional-radiative dielectronic recombination rate coefficient as a function of electron density as determined using the ADAS baseline approach i.e. the BBGP for dielectronic recombination. Density effects are negligible below ∼ 10 11 cm 3 s -1 . As the density increases then stepwise ionization reduces the peak rate coefficient by 20% at ∼ 10 13 cm 3 s -1 and by a nearly a factor of two at ∼ 10 15 cm 3 s -1 .

Sn 7+

We find that moving from Sn 11+ to Sn 7+ continues the isoelectronic trend noted on comparing and contrasting results for Sn 11+ to W 35+ . (The ground configurations 4d 3 and 4d 7 are equivalent from an angular momentum coupling perspective.) To illustrate: in figure 8 we show the configuration-average results for the total contribution from ∆n = 0 and ∆n = 1 promotions from the ground configuration. The ∆n = 0 dominate the high temperature peak and the ∆n = 1 only dominate at . Total dielectronic recombination rate coefficients for Sn 11+ as a function of electron density. Solid (red) curve: 10 10 cm -3 ; long-dashed (green) curve: 4.6 × 10 11 cm -3 ; short-dashed (blue) curve: 2.2 × 10 13 cm -3 ; dotted (purple) curve; 10 15 cm -3 . See text for details. All this work -see also Foster (2008). very low temperatures because there is no low temperature contribution associated with the ∆n = 0 promotion for this ion. Within the ∆n = 0 contributions the dipole are now completely dominant -see figure 9 -and the same is true for ∆n = 1 (not shown).

In figure 10 we compare our unmixed 4p -4d and 4d -4f results from autostructure with those from the configuration-average. The configurationaverage results are already around 80% larger than those from autostructure averaged-over the ground level. Averaging over all initial levels of the ground configuration increases the difference to about a factor of three.

In figure 11 we compare our configuration-mixed level-resolved autostructure results with the total rate coefficients from the BBGP and CA methods. The configuration-mixed autostructure result is only 5% smaller than the sum of the unmixed results (not shown) at the high temperature peak. The BBGP results are again somewhat smaller than our ones from autostructure. They also show a large low temperature contribution which somewhat improves the agreement with the autostructure results at the high temperature peak. The BBGP results are ∼30% smaller than the autostructure ones on making an allowance for this contribution. The low temperature variation of the rate coefficients is likely due to the 'random' positioning of near-threshold resonances. Our unmixed autostructure results in figure 10 show a comparably large contribution but peaked at a lower temperature. ) Energy (eV) CA AS (1) AS (CA) Sn 7+ Figure 10. Dielectronic recombination 4p -4d and 4d -4f contributions for Sn 7+ . Solid (red) curves: configuration-average (CA); long-dashed (green) curves: autostructure averaged over the initial ground level only; short-dashed (blue) curves: autostructure averaged over all levels of the ground configuration. For each pair of curves the lower corresponds to the 4p -4d contribution and the upper one the 4d -4f contribution. See text for details. All this work.

The configuration-average results have a negligible low temperature contribution and are nearly a factor of two larger than the autostructure results at the high temperature peak and are only marginal improvement over the results of the Burgess GF here.

Sn 8+ -Sn 10+

In figures 12, 13 and 14 we compare our configuration-mixed level-resolved autostructure results with the total rate coefficients from the BBGP method for Sn 8+ , Sn 9+ and Sn 10+ respectively. We compare also with CA results obtained from including just the 4p-4d and 4d-4f promotions now. These are the same promotions as included in the autostructure and Burgess GF results.

Sn 8+

The autostructure, BBGP and CA results all have similar low temperature contributions. The BBGP results lie 40% below, the CA 55% above and the GF 80% above the autostructure ones.

Sn 9+

The very large low temperature contribution in the autostructure results pushes the high temperature peak (in as much as there is one) close to that of the Burgess GF. We estimate the 'low temperature' contribution to be nearly half of the total at 50 eV. If we attempt to factor that out then the BBGP results are roughly 45% lower and the CA 45% higher -but clearly this is roughly. The Burgess GF results are again only a little larger than the CA ones here.

Sn 10+

The pattern of results here is similar to those just discussed for Sn 9+ . The BBGP results are roughly 15% lower and the CA 50% higher than the autostructure ones if we attempt to compensate for the 'low temperature' contribution at 50 eV. We show also the results obtained from the Flexible Atomic Code (FAC) [5]. They have a modest low temperature contribution which is comparable to the CA one. The FAC results are about 30% lower than the 'compensated' autostructure ones. The FAC calculations should be the closest to the autostructure ones. The 30% difference is perhaps a little large given the insensitivity to atomic structure or at least configuration-mixing. The FAC approach [6] solves the Dirac equation self-consistently using a unique model potential which is basically a weighted-average of the configuration-average potentials 'seen' by each orbital i.e. excluding self-interaction. This maintains orthogonality of the orbitals. We do not carry-out this second average but instead follow Cowan [START_REF] Cowan | The Theory of Atomic Structure and Spectra[END_REF] in neglecting the overlap contributions. Our use of the kappa-averaged relativistic wavefunctions should be sound for these outer-shell orbitals. A very accurate description of the atomic structure of Sn + has been obtained using only non-relativistic wavefunctions [25]. Total dielectronic recombination rate coefficients for Sn 8+ . Solid (red) curve: configuration-average; long-dashed (green) curve: autostructure averaged over the initial ground level; short-dashed (blue) curve: Burgess-Bethe General Program averaged over the initial ground level; dotted (purple) curve; Burgess General Formula. All this work.

Sn 12+ and Sn 13+

As we approach the shell boundary the complexity of the angular momentum coupling problem simplifies and becomes computationally less demanding. There may be additional complexities in describing the structure on breaking a closed-shell or the polarization of a single electron outside of a closed shell but they are beyond the scope of the present work. The shell boundary (Kr-like here) is important in plasma modelling since such ions are relatively more abundant over a wide temperature range.

Results (not shown) for the high-charge ions Sn 12+ and Sn 13+ show variable lowtemperature contributions similar to those seen for Sn 11+ . The Burgess GF results are about 30% larger than those from autostructure at the high temperature peak while the CA results are about 25% larger but those from the BBGP are nearly a factor of two smaller.

Sn 4+ -Sn 6+

The 'simple' picture of ∆n = 0 and ∆n = 1 transitions/promotions breaks down at low charges. Levels of the 4d (q-1) 5l configurations increasingly move below those of the 4d q-1 4f. This has two consequences. Rydberg states associated with the 4d-4f promotion now become increasingly energetically accessible to the 5s then 5p and finally the 5d continua. We expect the autostructure level-resolved autoionization rates associated with the 4d-4f promotion to dominate over 4f-5p and especially 4f-5s but there will be some suppression still. This is expected to be offset to a degree by the increasing direct contribution from ∆n = 1 promotions as they are no longer energetically accessible to the 4f continua.

The use of the CA method becomes increasingly problematic here. Levels of the 4d (q-1) 4f target/parent configuration become increasingly intermixed with those of the 4d (q-1) 5l. The CA method 'requires' that all levels of 4d q-1 4fnl be above or below 4d (q-1) 5s, 5p etc. This is impossible to satisfy in the high-n limit when the parents are intermixed. The 'high-n' limit becomes not so high at low charge. We illustrate this in the next section.

Sn 4+

The closed-subshell 4d 10 no longer admits the 4p -4d promotion and its consequential mixing with that arising from the 4d -4f promotion. It is a rather pure system. There is a single strong completely dominant ∆n = 0 radiative rate from the 4d 9 4f 1 P 1 level and this is the highest lying level in the configuration. The associated 4f-wavefunction is strongly term-dependent.

In figure 15 we compare the total dielectronic recombination rate coefficients for Sn 4+ obtained from the various methods. We present pairs of results for autostructure and configuration-average. One set is simply the result from including n = 4 target configurations (i.e. ∆n = 0 promotions). The second set includes n = 5 target configurations. The effect of the latter is quite complex.

We see that the n = 4 autostructure results lie substantially below the equivalent CA ones as well as the BBGP and Burgess GF. We interpret this as follows: Rydberg nl states attached to the 4d 9 4f 1 P 1 core can autoionize into the continuum Total dielectronic recombination rate coefficients for Sn 10+ . Solid (red) curve: configuration-average; long-dashed (green) curve: autostructure averaged over the initial ground level; short-dashed (blue) curve: Burgess-Bethe General Program averaged over the initial ground level; dotted (purple) curve; Burgess General Formula: all this work. Dot-dashed (cyan) curve, FAC results from [5].

of lower levels of the 4d 9 4f configuration for n > 5. This strongly suppresses the dielectronic recombination in this simple system. If we omit such autoionization channels from our autostructure calculation then the peak temperature rate coefficient increases by nearly a factor of two. Such non-dipole autoionizing transitions within a configuration are not described by either the BBGP or CA methods.

We can investigate the effect of the 4d 9 5l configurations which (for l = 0 -2) lie wholly (l = 0, 1) or partially (l = 2) below the 4d 9 4f. We find that their inclusion reduces the autostructure result for the peak temperature rate coefficient by only 5%. The reasons for this are twofold. The suppression of the DR via the 4d -4f promotion by autoionization into the n = 5 continua has to compete against the suppression via 4f -4f which is already included. Then there is a contribution to DR from the 4d -5p promotions which is only suppressed by autoionization into the 5s continuum. The two effects each affect the total rate coefficient by roughly 20-30% but are largely self-cancelling. Both effects are diminished in magnitude as the charge state increases.

The situation for the CA method is dramatically different. The inclusion of autoionization into the n = 5 continua reduces the 4d -4f peak temperature rate coefficient by about a factor of 12. The ∆n = 1 contribution is then comparable with the reduced ∆n = 0 and in contradistinction to the autostructure finding. This partially offsets the effect of the dramatic suppression of the 4d -4f contribution to the total. The net result is a factor of 7 reduction of the CA total peak temperature rate coefficient which is in turn about a factor of two smaller than the (nl = 5s -5d) autostructure result (not shown -see next.)

We turn now to term-dependence. The 4d 9 4f 1 P 1 → 4d 10 1 S 0 radiative rate from the n = 4 autostructure target is broadly comparable with the configurationaverage value: 3.3 × 10 11 s -1 vs 2.3 × 10 11 s -1 . We use Fischer's Hartree-Fock code [26] to obtain a term-dependent 4f 1 P wavefunction: the radiative rate is 6.6 × 10 10 s -1 . We can attempt to correct for term-dependence in the autostructure calculation by including the 4d 9 5f configuration. This results in a rate of 9.5 × 10 10 s -1 from the 4d 9 4f 1 P 1 . This factor 3 reduction translates into a 40% reduction in the peak temperature DR rate coefficient as the reduction of the 4f contribution due to termdependence is partially offset by the (enhanced) contribution from the 5f promotion.

There is no configuration mixing in the CA method and so no account is taken of term-dependence. The final term-dependent corrected autostructure peak temperature DR rate coefficient lies remarkably and likely coincidentally close (10%) to the CA one perhaps indicating the overall effect of unitarity on the totals.

3.5.2. Sn 5+ and Sn 6+ Results (not shown) for these two ions show a similar trend to those seen for Sn 7+ . The BBGP results differ by 5% and 20% respectively from the autostructure results at the high temperature peak while those of the Burgess GF and CA (without n = 5) are a factor of two larger. The low temperature results for these ions are reduced both in relative magnitude and variation compared to those already presented for Sn 7+ . If we include the 4d 9 5l configurations (l = 0 -2) we find that our autostructure result for the peak temperature rate coefficient changes by only 1%. (There are larger low temperature effects due to the change in position of near threshold resonances due to the change in the configuration interaction expansion.)

The situation for the CA improves with the increased charge but this is likely simply due to the importance of the fundamental effect itself decreasing as the standard ordering of 4f and 5l re-asserts itself. This reduces and then eliminates the n = 5 continua suppression effect on DR via the 4d -4f promotion and in turn causes the 4d -5p contribution to the recombination itself to be suppressed increasingly by autoionization into excited states. Inclusion of the n = 5 suppression of the ∆n = 0 contribution reduces the net high temperature peak total rate coefficient by a factor of 4 for Sn 5+ and 40% for Sn 6+ . This 40% reduction for Sn 6+ brings the CA result into good agreement (15%) with our autostructure result but it is coincidental. The contribution from ∆n = 1 promotions in their own right is now about 20% of the total for Sn 5+ .

We have not included the n = 5 suppression effect in the CA ∆n = 0 results presented for other ions. (Where a ∆n = 1 contribution is shown it was calculated separately.) We note here that the Sn 7+ total peak temperature rate coefficient is reduced by 25% if included. We recall that our autostructure result is reduced by nearly a factor of two if we choose to average over all levels of the ground configuration as opposed to just the ground level. The variation in the CA results should be seen in this perspective.

Fitting coefficients

It is convenient often for simple modelling purposes to fit the total dielectronic recombination rate coefficient to the following functional form:

α z (T ) = T -3/2 i c i exp -E i T (2)
where the E i are in the units of temperature T (e.g. eV) and the units of c i are then cm 3 s -1 [eV] 3/2 . In table 1 we present the fitting coefficients for our autostructure total dielectronic recombination rate coefficients for the initial ground level of Sn z+ (for z = 4 -13) which are based upon the functional form given by equation (2). The fits are accurate to better (often much better) than 5% over 1-1000 eV.

The total dielectronic recombination rate coefficients can be taken to be the total recombination rate coefficients. The contribution from radiative recombination is negligible over the given temperature range as is that from three-body recombination at the densities of interest to technical and magnetic fusion plasmas.

Recapitulation

We have made extensive calculations of dielectronic recombination rate coefficients for Sn z+ (z = 4 -13) ions using configuration-mixed Breit-Pauli, Burgess-Bethe and configuration-average methods. If we except the special case of the closed-subshell Sn 4+ ion then we can state some broad findings:

(i) dipole ∆n = 0 promotions dominate. This behaviour is in contrast to that found for W 35+ [9] (which is isoelectronic with Sn 11+ ) where both non-dipole and ∆n = 1 promotions made notable contributions. It is readily understood in terms of the z-scaling of the relevant energies and rates.

(ii) configuration-mixing between 4p 6 4d (q-1) 4f and 4p 5 4d (q+1) has a small effect on the total dielectronic recombination rate coefficient. This is despite the significant effect it has on individual radiative rates for example. Unitarity dominates.

(iii) the BBGP results are systematically lower than those obtained from autostructure. In some cases they are significantly lower (by a factor of two).

(iv) the CA results are systematically and significantly higher (by up to a factor of two) than the configuration mixed Breit-Pauli results obtained from autostructure. Their improvement over results obtained from the Burgess GF is marginal at best. This is in contrast to that found previously for W 35+ [9] and appears to be due to the lower residual charge here. Caveat: at the lowest charges the ∆n = 0 CA results are reduced significantly by including autoionization to the n = 5 continua. This is only partially offset by the ∆n = 1 contribution. The net effect is only a few percent effect in the fully level-resolved picture described by autostructure.

(v) the autostructure results differ significantly (by up to 70%) between averaging over just (the weight of) the ground level and all levels of the ground configuration.

Conclusions

The BBGP and CA methods should be used to determine the dominant promotions which contribute to the dielectronic recombination processes of interest for heavy species. They can be used also to 'top-up' small contributions. The BBGP method may be preferable since it describes N -electron configuration-mixing and it can resolve by initial level -this latter feature is especially important at low charge. But the CA method may still be required so as to assess the importance of nondipole promotions since the BBGP method requires externally sourced distorted wave collision strengths to describe them. A full level-resolved treatment appears to be necessary for the dominant contributions so as to obtain totals accurate to better than a factor of two compared to the more approximate BBGP and CA methods. The full configuration-mixed Breit-Pauli implementation within the autostructure code can describe such DR processes involving a half-open d-subshell. A complete set of such results for Tungsten ions is highly desirable. (Recall that the agreement between autostructure and CA for W 35+ was obtained by averaging-over all levels of the initial ground configuration [9].) The f-subshell will be the next challenge. This is necessary for describing DR processes in the Tungsten ions: W 13+ -W 27+ . It is likely that the BBGP and/or CA methods will have to play a greater role then.

N

  R Badnell, A Foster, D C Griffin, D Kilbane, M O 'Mullane, H P Summers To cite this version: N R Badnell, A Foster, D C Griffin, D Kilbane, M O 'Mullane, et al.

Figure 1 .

 1 Figure1. Total line strength emission in Sn 10+ as function of wavelength convoluted with a 0.01 nm FWHM Gaussian. The top spectrum is from the 4p -4d array and the middle from the 4d -4f array and where each has been calculated separately. The bottom spectrum is the configuration-mixed (4p 6 4d 3 4f with 4p 5 4d 5 ) sum of 4p -4d plus 4d -4f. See text for details. All this work.

Figure 2 .

 2 Figure 2. Configuration-average dielectronic recombination contributions for Sn 11+ . ∆n = 0 denotes the sum of the 4l -4l contributions and ∆n = 1 the sum of the 4l -5l . 'Tot.' denotes the sum of the ∆n = 0 plus ∆n = 1. See text for details. All this work.

Figure 3 .

 3 Figure 3. Configuration-average n = 4 -4 dielectronic recombination contributions for Sn 11+ . ∆n = 0 denotes the sum (l, l ) of the 4l-4l contributions which are shown individually. (The 4s -4d is off the bottom of the graph -i.e. negligible.) See text for details. All this work.

Figure 4 .

 4 Figure 4. Configuration-average dielectronic recombination contributions for Sn 11+ (a): n = 4p -5l and (b): n = 4d -5l. 'Sum' denotes the sum over l of the 4p, 4d -5l contributions which are shown individually. (The 4p, 4d -5g are off the bottom of the graph.) See text for details. All this work.

Figure 5 .

 5 Figure 5. Dielectronic recombination 4p -4d and 4d -4f contributions for Sn 11+ . Solid (red) curves: configuration-average (CA); long-dashed (green) curves: autostructure averaged over the initial ground level only; shortdashed (blue) curves: autostructure averaged over all levels of the ground configuration. For each pair of curves the upper corresponds to the 4p -4d contribution and the lower one the 4d -4f contribution. See text for details. All this work.

Figure 6 .

 6 Figure 6. Total dielectronic recombination rate coefficients for Sn 11+ . Solid (red) curve: configuration-average; long-dashed (green) curve: autostructure averaged over the initial ground level; short-dashed (blue) curve: Burgess-Bethe General Program averaged over the initial ground level; dotted (purple) curve; Burgess General Formula. All this work.

Figure 7

 7 Figure 7. Total dielectronic recombination rate coefficients for Sn 11+ as a function of electron density. Solid (red) curve: 10 10 cm -3 ; long-dashed (green) curve: 4.6 × 10 11 cm -3 ; short-dashed (blue) curve: 2.2 × 10 13 cm -3 ; dotted (purple) curve; 10 15 cm -3 . See text for details. All this work -see alsoFoster (2008).

Figure 8 .

 8 Figure 8. Configuration-average dielectronic recombination contributions for Sn 7+ . ∆n = 0 denotes the sum of the 4l -4l contributions and ∆n = 1 the sum of the 4l -5l . 'Tot.' denotes the sum of the ∆n = 0 plus ∆n = 1. See text for details. All this work.

Figure 9 .

 9 Figure 9. Configuration-average n = 4 -4 dielectronic recombination contributions for Sn 7+ . ∆n = 0 denotes the sum (l, l ) of the 4l -4l contributions which are shown individually. (The 4s -4l are off the bottom of the graph.)

Figure 11 .

 11 Figure 11. Total dielectronic recombination rate coefficients for Sn 7+ . Solid (red) curve: configuration-average; long-dashed (green) curve: autostructure averaged over the initial ground level; short-dashed (blue) curve: Burgess-Bethe General Program averaged over the initial ground level; dotted (purple) curve; Burgess General Formula. All this work.

Figure 12 .

 12 Figure 12. Total dielectronic recombination rate coefficients for Sn 8+ . Solid (red) curve: configuration-average; long-dashed (green) curve: autostructure averaged over the initial ground level; short-dashed (blue) curve: Burgess-Bethe General Program averaged over the initial ground level; dotted (purple) curve; Burgess General Formula. All this work.

Figure 13 .

 13 Figure 13. Total dielectronic recombination rate coefficients for Sn 9+ . Solid (red) curve: configuration-average; long-dashed (green) curve: autostructure averaged over the initial ground level; short-dashed (blue) curve: Burgess-Bethe General Program averaged over the initial ground level; dotted (purple) curve; Burgess General Formula. All this work.

Figure 14 .

 14 Figure 14. Total dielectronic recombination rate coefficients for Sn 10+ . Solid (red) curve: configuration-average; long-dashed (green) curve: autostructure averaged over the initial ground level; short-dashed (blue) curve: Burgess-Bethe General Program averaged over the initial ground level; dotted (purple) curve; Burgess General Formula: all this work. Dot-dashed (cyan) curve, FAC results from[5].

Figure 15 .

 15 Figure 15. Total dielectronic recombination rate coefficients for Sn 4+ : where there are pairs of curves the lower/upper are with/without n = 5 (see text). Solid (red) curves: configuration-average; long-dashed (green) curves: autostructure averaged over the initial ground level; short-dashed (blue) curve: Burgess-Bethe General Program averaged over the initial ground level; dotted (purple) curve; Burgess General Formula. All this work.

Table 1 .

 1 Total dielectronic recombination rate coefficient fitting coefficients c i (cm 3 s -1 [eV] 3/2 ) and E i (eV) for the initial ground level of Sn z+ .

	z	c 1	c 2	c 3	c 4	c 5
	z	E 1	E 2	E 3	E 4	E 5
	13	1.654(-1)	2.011( 0)	8.695( 0)	4.152(+ 1)	9.022(+ 1)
	12	1.717(-1)	1.413( 0)	6.970( 0)	4.062(+ 1)	8.971(+ 1)
	11	1.331(-1)	9.384(-1)	6.094( 0)	3.834(+ 1)	8.997(+ 1)
	10	8.116(-2)	9.333(-1)	4.000( 0)	4.602(+ 1)	9.643(+ 1)
	9	2.731(-1)	2.110( 0)	6.049( 0)	7.407(+ 1)	1.340(+ 2)
	8	7.158(-2)	5.526(-1)	3.232( 0)	2.054(+ 1)	8.255(+ 1)
	7	5.997(-2)	4.930(-1)	2.406( 0)	1.776(+ 1)	7.715(+ 1)
	6	1.958(-1)	1.385( 0)	1.275(+ 1)	6.674(+ 1)	1.195(+ 2)
	5	1.047(-1)	8.616(-1)	1.245(+ 1)	5.718(+ 1)	9.505(+ 1)
	4	2.493( 0)	1.611(+ 1)	5.145(+ 1)	6.909(+ 1)	2.379(+ 3)

a (m) denotes ×10 m .

[26] Froese Fischer C 1991 Comp. Phys. Commun. 64 369-98

Acknowledgments

This work was supported in part by a Euratom Framework 7 Support Action agreement with the University of Strathclyde.