PhD Marie-Noël Vercambre

ScD Francine Grodstein

MD, PhD Claudine Berr

ScD Jae H Kang

Mediterranean Diet and Cognitive Decline in Women with Cardiovascular Disease or Risk Factors

INTRODUCTION

Substantial evidence supports the role of cardiovascular disease (CVD) and coronary risk factors in both cognitive decline and dementia [START_REF] Battistin | Vascular Cognitive Disorder. A Biological and Clinical Overview[END_REF][START_REF] Reitz | A Summary Risk Score for the Prediction of Alzheimer Disease in Elderly Persons[END_REF]). Yet little is known regarding means of preserving cognitive health in this high-risk group, a group representing as much as 70% of those aged ≥ 65 years in the US [START_REF] Adams | Heart Disease and Stroke Statistics--2009 Update: A Report From the American Heart Association Statistics Committee and Stroke Statistics Subcommittee[END_REF]. Certainly, diet is known to modulate cardiovascular processes (4), and thus would appear a plausible behavioral approach to also reducing cognitive decline in those with vascular conditions. While some previous studies have focused on individual nutrients such as antioxidants, omega 3 fatty acids or B vitamins both in healthy participants and those with vascular risk, clinical trials of these agents have not shown either major vascular [START_REF] Sesso | Vitamins E and C in the prevention of cardiovascular disease in men: the Physicians' Health Study II randomized controlled trial[END_REF][START_REF] Kromhout | n-3 fatty acids and cardiovascular events after myocardial infarction[END_REF][START_REF] Galan | Effects of B vitamins and omega 3 fatty acids on cardiovascular diseases: a randomised placebo controlled trial[END_REF] or neuroprotective [START_REF] Eussen | Effect of oral vitamin B-12 with or without folic acid on cognitive function in older people with mild vitamin B-12 deficiency: a randomized, placebo-controlled trial[END_REF][START_REF] Kang | A randomized trial of vitamin E supplementation and cognitive function in women[END_REF][START_REF] Aisen | High-Dose B Vitamin Supplementation and Cognitive Decline in Alzheimer Disease[END_REF] effects in aging. In contrast, there may be additive and interactive effects between individual dietary components within a dietary pattern, and both randomized trials [START_REF] De Lorgeril | Mediterranean Diet, Traditional Risk Factors, and the Rate of Cardiovascular Complications After Myocardial Infarction : Final Report of the Lyon Diet Heart Study[END_REF][START_REF] Estruch | Effects of a Mediterranean-style diet on cardiovascular risk factors: a randomized trial[END_REF] and cohort studies [START_REF] Nunez-Cordoba | The Mediterranean diet and incidence of hypertension: the Seguimiento Universidad de Navarra (SUN) Study[END_REF][START_REF] Fung | Mediterranean Diet and Incidence of and Mortality From Coronary Heart Disease and Stroke in Women[END_REF][START_REF] Chrysohoou | The Mediterranean diet contributes to the preservation of left ventricular systolic function and to the long-term favorable prognosis of patients who have had an acute coronary event[END_REF] have reported reduced cardiovascular outcomes with a Mediterranean diet pattern in particular, which is characterized by high intakes of fruits and vegetables, fish, cereals, legumes, and monounsaturated fat, moderate alcohol, and low amounts of meat and dairy products [START_REF] Sofi | Accruing evidence on benefits of adherence to the Mediterranean diet on health: an updated systematic review and meta-analysis[END_REF]. Through modulation of vascular processes, Mediterranean diet may then potentially slow cognitive aging. However, there has been very limited prospective research on the Mediterranean diet and cognitive health [START_REF] Scarmeas | Mediterranean diet and risk for Alzheimer's disease[END_REF][START_REF] Feart | Adherence to a Mediterranean Diet, Cognitive Decline, and Risk of Dementia[END_REF][START_REF] Tangney | Adherence to a Mediterranean-type dietary pattern and cognitive decline in a community population[END_REF], and to our knowledge, no data in those with prevalent vascular disease or risk factors. In fact, even factors which may help maintain cognition in healthy older persons, may not be equally effective in those with existing vascular disease, as pathology in the later is likely more severe and potentially less tractable.

Thus, the potential effect of Mediterranean diet on cognitive decline was specifically evaluated in older persons with vascular disease, utilizing data from the cognitive ancillary study among the oldest participants in the Women"s Antioxidant Cardiovascular Study (WACS), a trial of secondary prevention of CVD.

METHODS

The Women"s Antioxidant Cardiovascular Study (WACS) began in 1995-1996 ("baseline")

among 8171 American women, as a 2x2x2 randomized placebo-controlled trial of vitamin E, vitamin C, and beta-carotene supplementation for secondary prevention of CVD [START_REF] Cook | A randomized factorial trial of vitamins C and E and beta carotene in the secondary prevention of cardiovascular events in women: results from the Women's Antioxidant Cardiovascular Study[END_REF]. Eligible participants were US female health professionals, aged ≥ 40 years, with prevalent CVD or ≥ 3 coronary risk factors (i.e., parental history of premature myocardial infarction (MI), diabetes, hypertension, hyperlipidemia, and body mass index ≥ 30 kg/m 2 ). Prevalent CVD included MI, stroke, revascularization procedures (i.e., percutaneous transluminal angioplasty, coronary artery bypass graft, carotid endarterectomy, or peripheral artery surgery), symptomatic angina pectoris, or transient cerebral ischemia. In 1998, a fourth arm for B vitamin supplementation (combined folic acid, vitamin B6, and vitamin B12) was added among 5442 women [START_REF] Albert | Effect of folic acid and B vitamins on risk of cardiovascular events and total mortality among women at high risk for cardiovascular disease: a randomized trial[END_REF]. Participants were 94.0% Caucasian and 3.3% African-American. Until the scheduled end in 2005, they completed annual questionnaires on compliance, side effects, health, lifestyle and clinical endpoints. This study was approved by the institutional review board of Brigham and Women"s Hospital, Boston, MA. Each participant gave a written informed consent. None of the supplements were found to reduce cardiovascular disease recurrence [START_REF] Cook | A randomized factorial trial of vitamins C and E and beta carotene in the secondary prevention of cardiovascular events in women: results from the Women's Antioxidant Cardiovascular Study[END_REF][START_REF] Albert | Effect of folic acid and B vitamins on risk of cardiovascular events and total mortality among women at high risk for cardiovascular disease: a randomized trial[END_REF] or influence cognitive decline [START_REF] Kang | A trial of B vitamins and cognitive function among women at high risk of cardiovascular disease[END_REF][START_REF] Kang | Vitamin C, Beta Carotene, and Cognitive Function Among Women With or at Risk of Cardiovascular Disease: The Women's Antioxidant and Cardiovascular Study[END_REF].

Cognitive subcohort and study population

From 1998-2000, cognitive function were assessed using telephone interviews among participants then aged ≥ 65 years. Of the 3170 eligible women, 190 were unreachable, 156 declined participation, and 2824 (95% of contacted women) completed the initial telephone cognitive assessment. These women received three follow-up assessments at two-year intervals until 2005; 93% completed at least two cognitive assessments, and 81% completed at least three among four. For the fourth assessment, 24% of participants were not contacted as only a short interval had passed between their third interview and the trial end. The mean time from the initial to the last cognitive assessment was 5.4 years (range 4.1-6.1 years).

For this study on diet and cognitive decline, 320 participants among the 2824 with initial cognitive assessment were excluded because of incomplete dietary information. Thus, the analysis sample for the present study was comprised of 2504 women.

Cognitive Assessment

The telephone interview consisted of five cognitive tests. Global cognition was evaluated with the Telephone Interview of Cognitive Status (TICS) [START_REF] Brandt | The telephone Interview for Cognitive Status[END_REF] (range 0 to 41 points), a telephone adaptation of the Mini-Mental State Examination (MMSE). Verbal memory was assessed with the TICS 10-word list (immediate and delayed recalls) and the East Boston Memory Test (immediate and delayed recalls) [START_REF] Scherr | Correlates of cognitive function in an elderly community population[END_REF]. A test of category fluency [START_REF] Morris | The Consortium to Establish a Registry for Alzheimer's Disease (CERAD). Part I. Clinical and neuropsychological assessment of Alzheimer's disease[END_REF], in which women were asked to name as many animals as possible in one minute, was also administered.

The primary outcome was the rate of change from the initial through the last assessment in a global composite score, computed as the mean of the z-scores from all cognitive tests. The secondary outcomes were the changes in TICS score, verbal memory composite score (mean of the z-scores from the immediate and delayed recalls of both TICS-10 word list and the East Boston Memory Test), and category fluency score. Verbal memory is among the best predictors of Alzheimer disease [START_REF] Small | The course of cognitive impairment in preclinical Alzheimer disease: three-and 6-year follow-up of a populationbased sample[END_REF], and category fluency partly measures executive function, which is associated with vascular disease. The means of the z-scores from the available relevant tests were used to derive the composite scores for participants who did not complete all tests (only 0.5% of participants). Extensive evidence supports the face and clinical validity of this telephone cognitive instrument as previously developed [START_REF] Vercambre | Physical activity and cognition in women with vascular conditions[END_REF].

Mediterranean diet

The Willett semi-quantitative food frequency questionnaire was administered at WACS baseline. The questionnaire asked about the usual consumption during the past year (nine possible response categories) of 116 food items with a specified portion size. It has been extensively studied for validity [START_REF] Willett | Reproducibility and validity of a semiquantitative food frequency questionnaire[END_REF]. The mean time from the food frequency questionnaire to the initial cognitive assessment was 3.5 years (range 3.1-4.4 years). This lag period likely has some benefits. First, since diet was assessed at somewhat younger ages, it is likely less influenced by "reverse" causationthat is, changes in diet due to underlying cognitive status. Moreover, biologically, diet at more remote timepoints is probably more relevant to brain health than more immediate diet, as cognitive decline develops over many years [START_REF] Small | The course of cognitive impairment in preclinical Alzheimer disease: three-and 6-year follow-up of a populationbased sample[END_REF].

A widely-used method based on a 0 to 9-point scale was implemented to evaluate the degree of individual adherence to a Mediterranean diet pattern [START_REF] Trichopoulou | Diet and overall survival in elderly people[END_REF][START_REF] Trichopoulou | Adherence to a Mediterranean diet and survival in a Greek population[END_REF][START_REF] Trichopoulou | Modified Mediterranean diet and survival: EPIC-elderly prospective cohort study[END_REF], where higher scores indicated stricter adherence. First, dietary intakes were energy-adjusted using the residual method. Then, for each of nine major components of the Mediterranean diet (vegetables, legumes, fruits, cereals, fish, meat, dairy products, ratio of monounsaturated to saturated fat and alcohol), a value of zero or one was assigned, according to presumed level of health benefit. Specifically, women whose consumption of vegetables, legumes, fruits, cereals, fish, and monounsaturated to saturated fat ratio was above the median consumption in the study population were assigned a value of one, and a value of zero otherwise. In contrast, women whose consumption of meat and dairy products was above the median consumption were assigned a value of zero and a value of one otherwise. Finally, for alcohol intake, women with mild-to-moderate alcohol consumption (from 5 to 25 g of alcohol per day) were assigned a value of one and a value of zero otherwise [START_REF] Trichopoulou | Adherence to a Mediterranean diet and survival in a Greek population[END_REF][START_REF] Trichopoulou | Modified Mediterranean diet and survival: EPIC-elderly prospective cohort study[END_REF]. The Mediterranean diet score was generated by adding all the binary component scores. Its mean was 4.2 (SD, 1.8) and its distribution was normal. This sum was then divided into three categories (low, middle and high), broadly corresponding to tertiles: 0-3; 4-5; 6-9. Mean intakes of consumption for the nine diet components are presented in Table 1.

In secondary analysis, an alternate definition for the Mediterranean diet adherence was considered through a 0-55 point diet score used in a recent study of Mediterranean diet and cognitive decline [START_REF] Tangney | Adherence to a Mediterranean-type dietary pattern and cognitive decline in a community population[END_REF]. This adaptation from the score described by Panagiotakos et al. [START_REF] Panagiotakos | Adherence to the Mediterranean food pattern predicts the prevalence of hypertension, hypercholesterolemia, diabetes and obesity, among healthy adults; the accuracy of the MedDietScore[END_REF] was based on reported intakes of alcohol and 10 food groups: seven components consistent with the Mediterranean diet (nonrefined cereals, potatoes, fruits, vegetables, legumes / nuts / beans, fish, olive oil) and three "inconsistent" components (red and processed meats, poultry, full-fat cheese and other dairy). The scores for each component ranged from 1 to 5 based on the level of consumption frequency observed at the population level.

To further assess the robustness of the results and susceptibility to recent changes in diet, subgroup analyses were conducted only among women who reported at WACS baseline that their diet changed very little in the past five years (n=847).

Covariates

Annual questionnaires to WACS participants were used to obtain information on a wide variety of potential confounders including sociodemographic factors, health, medications and lifestyle factors plausibly linked with both cognitive decline and dietary habits. Variables introduced in the models as adjustment variables included: age (years), education (licensed practical nurse, vocational nurse or associate"s degree; registered nurse degree or bachelor"s degree; master"s or doctoral degree), energy from diet (quartiles), marital status (married, divorced, widowed, single), physical activity (quartiles of weekly calories expended from exercise and climbing the stairs), use of multivitamin supplements (no, yes), smoking status (never, past, current), body mass index (quartiles), postmenopausal hormone therapy use (never, past, current), aspirin use exceeding 10 days in the previous month (no, yes), non-steroidal antiinflammatory drug use exceeding 10 days in the previous month (no, yes), history of depression (no, yes), cardiovascular profile at baseline (myocardial infarction, stroke, revascularization procedures, symptomatic angina pectoris, transient cerebral ischemia, no clinical disease), diabetes (no, yes), hypertension (no, yes on pharmaceutical treatment, yes without pharmaceutical treatment), hyperlipidemia (no, yes on pharmaceutical treatment, yes without pharmaceutical treatment), and randomization assignment for vitamin E (placebo, active), vitamin C (placebo, active), beta-carotene (placebo, active), and folate (not assigned, placebo, active). In numerous validation studies in populations similar to WACS, self-reported covariate data from studies of health professionals have proven to be highly valid.

Statistical Analysis

First, participants" age and age-standardized characteristics at dietary assessment were compared across tertiles of Mediterranean diet score. Homogeneity was tested using p-values from Chi 2 test or variance analysis depending on the covariate type (categorical or numeric).

Then, general linear models for repeated measures with random intercepts and slopes were used to estimate the association of Mediterranean diet score with the annual rate of cognitive change. The longitudinal correlation in scores within subject was incorporated into the models using an unstructured covariance matrix. Linear trends across level of Mediterranean diet adherence were tested by assigning the median value to each of the three categories of the Mediterranean diet score as a continuous ordinal variable. Wald tests were used for statistical testing. All models were fitted by maximum likelihood method using the SAS software (SAS release 9.1, SAS Institute Inc., Cary, NC).

Basic models included adjustment for age at initial cognitive assessment, educational attainment, and energy from diet. Full multivariable models further adjusted for WACS randomization assignments, and numerous lifestyle and health variables (as described above).

Secondary analysis also adjusted for incident vascular events during follow-up.

Given the possibility that age, education, depression, cardiovascular profile at baseline or cognitive status at initial assessment could modify the association between Mediterranean diet and cognitive decline, interaction terms for these variables and Mediterranean diet score were evaluated, and stratified analyses by these variables were conducted as well. Potential effect modifiers were defined as follows: age at first cognitive assessment (age 65 -72

RESULTS

In the study population, mean intakes of vegetables and legumes nearly doubled across tertiles of Mediterranean diet score and mean intakes of fruit, fish and grains increased by nearly 50% (Table 1). Proportion of mild-to-moderate alcohol consumers was 9% in the first tertile, 17% in the second and 30% in the third tertile of Mediterranean diet score.

As expected, stricter adherence to the Mediterranean diet was linked with higher education, lower body mass index and generally healthier behaviors (e.g., greater physical activity, less smoking) (Table 2). Importantly, adherence to Mediterranean diet was not clearly associated with cardiovascular profile at WACS baseline (i.e., history of clinical MI, stroke, revascularization surgery, angina, or transient ischemic attack; history of diabetes; history of hypertension), suggesting that the severity of cardiovascular disease at baseline was not a major confounder in the present study.

Mediterranean diet and cognitive change

In both the basic-adjusted and multivariable-adjusted models, no significant differences were observed across the categories of the Mediterranean diet score in the mean annual rate of cognitive change for the global composite, TICS, verbal memory, or category fluency scores (Table 3). For example, the mean multivariable-adjusted difference (95% confidence interval (CI)) in rates of change in the global composite score was 0.01 (-0.01, 0.02) between the second tertile and the first tertile of Mediterranean diet score, and 0.00 (-0.02, 0.01) between the top tertile and the first tertile (p-value for trend: 0.88).

Results were not substantially altered when models were controlled for incident major cardiovascular events during follow-up. In addition, results were consistently null when the Mediterranean diet score was considered as a continuous variable (e.g., for the global composite score, mean difference per additional score point was -0.001 (p-value for trend=0.40). Also, no relations were observed with the alternate Mediterranean diet score: p-values for trend in cognitive change across tertiles were 0.58 for the global composite score, 0.25 for the TICS, 0.39 for the verbal memory score and 0.52 for the category fluency score.

In the 847 women who reported stable diets (34% of the whole sample), effect estimates were virtually identical to those from the entire cohort, and there were no statistically significant associations between Mediterranean diet score and cognitive decline (e.g., for global composite score, mean difference (95% CI) between top and bottom tertiles of Mediterranean diet score was 0.00 (-0.03, 0.03), p-value for trend: 0.95).

Effect modification and stratified analyses

When investigating whether the associations with Mediterranean diet on cognitive decline may differ by baseline age, education, depression history at baseline, cardiovascular profile at baseline or cognitive score at first assessment, no significant interactions were found. Similarly, stratified analyses revealed no quantitative or qualitative differences between associations across strata of any of these variables (data not shown in table).

DISCUSSION

In this large prospective study of older women with vascular disease or at high vascular risk, no association was observed between adherence to the Mediterranean diet and cognitive decline over 5 years of follow-up. The absence of association was consistent across various cognitive outcomes, different definitions of Mediterranean diet scores and various strata of the population. There was no suggestion of an association among women who reported that their diet had been stable for at least the past 5 years.

Of three large studies which have examined the association between Mediterranean diet and cognitive status in generally healthy participants, the two American studies have reported less cognitive decline [START_REF] Tangney | Adherence to a Mediterranean-type dietary pattern and cognitive decline in a community population[END_REF] and lower risk of incident mild cognitive impairment [START_REF] Scarmeas | Mediterranean diet and mild cognitive impairment[END_REF] and incident Alzheimer disease [START_REF] Scarmeas | Mediterranean diet and risk for Alzheimer's disease[END_REF] with greater adherence to a Mediterranean diet. However, in the French study [START_REF] Feart | Adherence to a Mediterranean Diet, Cognitive Decline, and Risk of Dementia[END_REF], higher adherence to the Mediterranean diet was only weakly associated with slower decline in the MMSE and not consistently with other cognitive tests, and adherence was not related to risk of dementia. Thus, available results on the potential effect of Mediterranean diet in cognitive aging come from a limited number of inconsistent studies. What should be particularly underlined is the great variability across these studies in terms of "absolute" adherence to the Mediterranean diet : based on a comparison of the reported intakes of the nine components for the various categories of adherence, reference group in the present study population had higher intakes of vegetables, cereals and a much higher monounsaturated / saturated fat ratio than the other studies from the US, and was quite similar to that in the French cohort, which also found null associations with decline in cognition (except for MMSE) and with dementia. Moreover, it is possible that prevention is even more challenging in those with prevalent vascular disease or risk factors, as in WACS.

Limitations of this study should be discussed. The WACS dietary data were collected at trial baseline with a self-administered food frequency questionnaire, which although widely used in epidemiologic studies [START_REF] Willett | Reproducibility and validity of a semiquantitative food frequency questionnaire[END_REF], is subject to misclassification of exposures. This misclassification is highly likely to be non-differential because diet was assessed 3.5 years before cognitive function assessment; such non-differential misclassification would bias results towards the null and may explain the observed lack of association between Mediterranean diet and cognitive decline. However, other studies of diet and cognitive decline using this food frequency questionnaire in populations of health professionals have reported strong relations between diet and cognition [START_REF] Kang | Fruit and vegetable consumption and cognitive decline in aging women[END_REF][START_REF] Devore | Dietary fat intake and cognitive decline in women with type 2 diabetes[END_REF], indicating that misclassification is likely not a primary factor in the results here. Moreover, the two studies which have found associations between the Mediterranean diet and cognitive health [START_REF] Tangney | Adherence to a Mediterranean-type dietary pattern and cognitive decline in a community population[END_REF][START_REF] Scarmeas | Mediterranean diet, Alzheimer disease, and vascular mediation[END_REF] both used the same food frequency questionnaire to collect their dietary data. More importantly, the single assessment of dietary habits in late adulthood may not reflect the long-term dietary intakes of participants (in particular at midlife), which may be more etiologically relevant. This may be particularly true in this population of women at high cardiovascular risk, who were likely encouraged to change their diet at some point. To address this, the analyses were carefully adjusted for participants" cardiovascular profiles at WACS baseline. Indeed, those with clinical disease rather than only risk factors may be more motivated to change their diet. Also, subgroup analyses were conducted only among women who reported at baseline that their diet changed very little in the past five years. All of these approaches yielded findings very similar to the main analyses. Moreover, long-term dietary change is hard to achieve even in those with vascular risk [START_REF] Erkkila | Concordance of diet with the recommended cholesterol lowering diet in patients with coronary heart disease[END_REF][START_REF] Ma | Dietary quality 1 year after diagnosis of coronary heart disease[END_REF] and no evidence was found in the data that Mediterranean diet adherence was related to level of clinical cardiovascular disease, suggesting that adoption of a Mediterranean diet was likely not determined or highly influenced by underlying health. In addition, the study population showed fairly wide ranges in reported intake of many components in the Mediterranean diet (e.g., a doubling or near doubling in vegetables and legumes intakes from bottom to top tertile and 50% increase in fruit and grain intake), thus there was large variability in the Mediterranean diet across participants.

Concerning the cognitive evaluation, a telephone assessment might lack validity. However, both reliability and validity studies of the telephone instrument (compared to in-person interviews) have provided convincing evidence of its utility to evaluate cognitive function in epidemiologic setting. Moreover, using the same or a similar telephone battery, significant associations with cognitive aging have been found with a large number of risk factors, including dietary variables establishing that the telephone instrument validly identifies risk factors for cognitive decline. Another limitation is that participants were mostly Caucasian, which precludes extending the findings to other ethnic minorities. Finally, residual confounding is always a possibility in observational studies, although the analyses were carefully adjusted on various potential confounders for the association between diet and cognitive change.

In summary, in a large sample of community-dwelling women aged ≥ 65 years with preexisting cardiovascular disease or risk factors, no associations were observed between Mediterranean diet adherence and 5-year cognitive decline. Given the limited number of studies of Mediterranean diet and cognitive health, potential effects of Mediterranean diet in brain aging and dementia requires further evaluation, both in generally healthy participants and in those at higher risk of cognitive decline. To have a better view in the matter, it would be interesting to test various approaches for the Mediterranean diet, not only population-dependant approaches of adherence (such as the 0-9 point scale used here), but also individual ranking based on absolute dietary intakes.

  [n=1261] versus age 73 -91 [n=1243]); highest attained education (licensed practical nurse, vocational nurse, associate"s or registered nurse degree [n=1802] versus bachelor"s or master"s or doctoral degree [n=702]), history of depression at baseline (no [n=2128] versus yes |n=376]), baseline cardiovascular status (CVD event [n=1882] versus risk factors only [n=622]) and cognitive status at first assessment (global cognitive score ≤ median [n=1252] versus global cognitive score > median [n=1252]).


Roman B G Kramer

V S Egorov

V A Gasparov

A G M Jansen

W Joss

Condon Domain Phase Diagram for Silver

Keywords: numbers: 75.45.+j, 71.70.Di, 75.60.-d

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I. INTRODUCTION

The formation of dia-and paramagnetic domains has been predicted by Condon 1 to occur in non-magnetic pure metals by considering the collective interaction between the electrons on Landau-quantized energy levels in the de Haas-van Alphen (dHvA) effect. These domains corresponding to an inhomogeneous magnetization are commonly called Condon domains. The domain formation results from a self-consistent treatment of the oscillating dHvA magnetization M due to the orbital quantization of the electronic system in the total magnetic induction B = µ 0 (H + M ), where M = M (B) depends on the total induction B in an applied magnetic field H. Following the Pippard-Shoenberg concept of magnetic interaction 2,[START_REF] Shoenberg | Magnetic oscillations in metals[END_REF] where the electrons experience the influence of the magnetic field induced magnetization of all neighboring electrons, a thermodynamic instability arises when the dHvA amplitude becomes large enough, i.e., the differential susceptibility

χ = µ 0 ∂M ∂B > 1. (1)
This condition corresponds to the situation where the amplitude of the magnetization amplitude becomes comparable to the magnetic field period of the dHvA effect. For sufficiently strong magnetization amplitudes, this instability condition, rewritten like µ 0 ∂H/∂B = 1 -χ < 0, occurs in a certain field interval within the paramagnetic part of each dHvA cycle. In these field intervals, where µ 0 ∂H/∂B < 0, the induction as function of the applied field B(H) is multi-valued, like the van der Waals isotherm for a real gas. The system avoids this instability in the same way as the real gas. For an infinite long rod-like sample (demagnetization factor n = 0), the induction B undergoes a discontinuous transition between the two stable states with the induction B 1 and B 2 at a certain applied field H, like the liquid-gas specific volumes change discontinuously at the equilibrium vapor pressure. Both stable states B 1 and B 2 correspond to the same free energy and the inductions in the instability interval (B 1 , B 2 ) are never realized. Figure 1(a) shows schematically the magnetization energy E mag = 1 2µ0 (Bµ 0 H) 2 for n = 0 and the oscillating dHvA energy described by the LK-formula E osc = a cos(2πF/B) to its simplest approximation with a the oscillation amplitude and F the dHvA frequency. The sum of both energies as function of B is shown for three different magnetic fields in Fig. 1(b)-(d). Usually, there is only one minimum in the total energy for a given applied magnetic field and the system will assume this value of B. However if the curvature of the E mag is smaller than the curvature of E osc two minima coexist at an applied field H 1 [Fig. 1(c)] and the induction will jump discontinuously from the value B 1 to B 2 when sweeping the magnetic field through H 1 .

For a plate-like sample, oriented normal to the applied magnetic field H (n = 1), the boundary condition

B = µ 0 [H + (1 -n)M ] = µ 0 H is required even in the interval B 1 < µ 0 H < B 2 .
Therefore, the induction B can not change discontinuously and a homogeneous state is no longer possible. The plate breaks up into domains of opposite magnetization. The volume fractions of the domains with the respective inductions B 1 and B 2 are adjusted in a way that for the average induction of the sample B = µ 0 H is fulfilled 1 . The regions with B 1 < µ 0 H are diamagnetic, those with B 2 > µ 0 H are paramagnetic. The domain walls between the phases B 1 and B 2 run parallel to H across the plate. In contrast to 1: (Color online) (a) Schematic representation of the magnetization energy E mag and the dHvA-energy Eosc as function of B for n = 0. (b)-(d) Sum of these energies for three applied magnetic fields H0 < H1 < H2. If the curvature of the parabola is smaller than the curvature of the oscillating energy two minima coexist at H 1. This leads to discontinuous jump of the induction when sweeping the magnetic field through H1. magnetic domains in common ferro-and antiferromagnetism, Condon domains do not have their origin in the interaction of electrons via their spin moments, but via their orbital motion.

d) c) E B µ 0 H 0 E mag +E osc B b) a) E B B 2 B 1 E mag +E osc µ 0 H 1 E B µ 0 H 2 E mag +E osc B E B E mag E osc µ 0 H 0 FIG.
For a sample shape with intermediate demagnetization factor, 0 < n < 1, the above mentioned interval of magnetic field B 1 < µ 0 H < B 2 with the occurrence of the instability will be reduced compared to the plate-like sample with the field range of this interval proportional to n. Therefore, samples of arbitrary shape will still show the nonuniform domain state with the same dia-and paramagnetic phases, B 1 and B 2 , whose domain structure might however be more complex.

Besides the analogy with the van der Waals gas, there is a close analogy of the CDS with the intermediate state of type-I superconductors, where the same boundary condition of a magnetic field applied to a sample of nonzero demagnetization factor leads to the formation of alternating domains in the normal and superconducting state. Condon domains, however, have the unique feature that the transition between the uniform and the inhomogeneous domain state occurs periodically in subsequent dHvA oscillations.

Equation 1 defines the boundary between the uniform and the Condon domain state. The resulting CDS phase diagram in the (H, T ) plane can be predicted by means of the Lifshitz-Kosevich (LK) formula for the oscillatory magnetization of the dHvA signal using the Fermi surface parameters, like the curvature A ′′ = ∂ 2 A/∂k 2 of the Fermi surface cross section A and the effective mass m * , and the Dingle temperature T D as a parameter for the impurity-scattering damping of the signal [START_REF] Gordon | [END_REF] .

Up to now, Condon domains have been observed by different experimental methods: by NMR 5 , muon spin rotation (µSR) spectroscopy 6,7 and, more recently, by local Hall probes 8 . All experimental observations have in common that two distinct inductions B 1 and B 2 or an induction splitting δB = B 2 -B 1 are measured at a given applied field H and temperature T . However, these measurements yielded only a few points well inside the (H, T ) diagram where Condon domains exist that could be compared with the theoretically predicted diagram. As a consequence, for example, the data on beryllium obtained by µSR required new phase diagram calculations with a modified LK-formula for the susceptibility 9 . The exact determination of the CDS phase boundary, where δB approaches zero, is difficult and time-consuming with a difference measurement of B 1 and B 2 10 . It was shown recently that a small hysteresis occurs in the measured dHvA signal upon passing the CDS phase boundary 11 . Due to the irreversible magnetization, an extremely nonlinear response to a small modulation field arises in standard ac susceptibility measurements. The out-of-phase signal and the third harmonic of the pickup voltage rise steeply at the transition point to the CDS. The threshold character of these quantities offers therefore a possibility to measure a Condon domain phase diagram. One should note that the third harmonic of the susceptibility is commonly used as a very sensitive tool to detect phase boundaries also of other systems like e.g. the vortex-glass transition in superconductors 12 .

In this article we determine the Condon domain phase diagram for silver using the third harmonic of the ac susceptibility for the detection of the nonlinear magnetic response. It was shown earlier 13 that detailed calculations of the magnetoquantum oscillations in silver based on the LK-formula are in good agreement with experimental dHvA data up to 10 T. This is certainly due to the nearly spherical Fermi surface of silver. Expecting a good agreement with the theoretically determined CDS phase diagram, we applied to silver this first detailed determination of the phase diagram.

II. EXPERIMENTAL

The measurements were performed on a high quality silver single crystal of 4.1 × 2.1 × 1.0 mm 3 . The sample was cut from the same piece than the sample used for the direct observation of Condon domains using local Hall probe detection 8 . The sample preparation is described in detail elsewhere 14,15 . The sample has a residual resistance ratio R 300 K /R 4.2 K = 1.6×10 4 , measured by the contactless Zernov-Sharvin method 16 . The high quality of the sample results in a very low Dingle temperature, which was estimated from standard dHvA analysis to be about T D = 0.2 K yielding an electronic mean free path of about 0.8 mm.

A standard ac modulation method with a compensated pickup coil system was used. Both pickup coils are identical and consist of about 400 turns. A long coil wound by a copper wire produced the modulation field with variable amplitude at frequencies of 20 -200 Hz. The pickup voltage was simultaneously measured by two lock-in amplifiers on the first and on higher harmonics. The measurements were performed in a superconducting coil up to 16 T as well as in a resistive coil up to 28 T at temperatures of 1.3 -4.2 K. The long side of the sample was parallel to the [100]-axis of the single crystal and was slightly tilted (∼ 5 deg) with respect to the direction of the applied magnetic field so that only the dHvA frequency from the "belly" orbit of 47300 T existed in the frequency spectrum.

The method of nonlinear detection, we use here, is applied to determine for the first time a CDS phase diagram over a broad range of temperatures and magnetic fields. Therefore, we will present carefully the technical details of the measurements in order to show the robustness of the phase diagram determination with respect to changing experimental parameters and measurement conditions.

III. EVIDENCE OF HYSTERESIS IN SILVER

The employed method to determine a Condon domain phase diagram is based on the appearance of hysteresis in the CDS which was first discovered on beryllium 11 . Hysteresis appears at the phase transition to the CDS and this results in some radical changes in the response to an ac modulation field. In the following, we will show that the characteristic nonlinear features in the ac response, as found in beryllium, are also observed in silver.

In presence of hysteresis the amplitude of the susceptibility, normalized on the modulation level, depends on the modulation amplitude. This is expected to occur when the modulation level is of the order of the hysteresis loop width. The schematic representation of a hysteresis loop in Fig. 2 explains this nonlinear response to an ac field modulation. As a result, after the transition to the CDS, the positive (paramagnetic) part of the susceptibility turns out to be reduced. From a comparison of two normalized susceptibilities, one measured with high and the other with low modulation level, we can in principle find where the amplitude reduction starts and thereby the transition point to the CDS.

Figure 2 shows as well that the response to a sinusoidal field modulation becomes window shaped and is slightly shifted in phase with respect to the input. Therefore, both the third harmonic and the out-of-phase signal of the pickup voltage increase steeply when the CDS phase boundary is crossed 11 . This threshold behavior offers M H FIG. 2: (Color online) Schematic representation of two minor hysteresis loops. The ac response upon field modulation is window-like and slightly shifted in phase. The normalized response (apparent susceptibility) decreases sharply when the modulation amplitude becomes of the order of the hysteresis loop width.

a simple way to determine the transition point of the CDS. The major advantage of third harmonic and out-ofphase part measurements is that only one magnetic field or temperature sweep through the transition is needed.

Figures 3 and4 show the above discussed nonlinear features in the pickup signal at constant temperature T = 2.7 K measured in the superconducting coil. Figure 3(a) shows two traces of the normalized pickup voltage, i.e. the susceptibility, obtained in the same conditions with 1.0 G and 0.2 G modulation amplitude at 160 Hz modulation frequency. In principle, both modulation levels are small enough compared to the dHvA period of about 20 G at 10 T that identical traces are expected for the susceptibility. The expanded view around 8 T, which is outside the CDS, shows that the normalized signals are indeed identical. For higher fields, on the other hand, the upper part of the susceptibility waveform, measured with the smaller modulation level, is reduced. The expanded view around 13.5 T shows that the signals are identical except for the positive part of the oscillation. This implies that at this part of the dHvA oscillation the magnetization is irreversible and there is a small hysteresis loop. The width of the hysteresis loop is of the order of 0.2 G.

A similar decrease of the normalized response was observed earlier 17 on silver at low temperatures. Unfortunately, because of the absence of the experimental parameters, this study can be compared only qualitatively with our data.

The magnetic field where the normalized pickup voltages start to differ between low and high modulation level is marked approximately by an arrow in Fig. 3(a). We obtain for the critical magnetic field µ 0 H c1 = 10.0 T.

Figure 3(b) shows the behavior of the third harmonic which was simultaneously measured with the first harmonic response in Fig. 3(a) for 0.2 G modulation am- plitude. For magnetic fields lower than the critical field there is only noise. At the transition to the CDS hysteresis arises and the third harmonic increases very steeply. This is nicely seen in the respective expanded views. The critical field of the CDS phase boundary can be obtained as the intersection of the two straight lines shown in Fig. 3(b). Here, the critical field is found as

µ 0 H c2 = 9.6 T.
The amplitude of the third harmonic is expected to go to zero in each diamagnetic part of the dHvA period because the sample magnetization is here homogeneous and without hysteresis. The presented behavior in Fig. 3(b) does not go to zero exactly which is certainly the result of a small rectification effect or, what is the same, the result of phase smearing of the oscillation signal. The homogeneity of the coil is about 30 ppm in a sphere with 1 cm diameter which may result in a field inhomogeneity of about 1 G in the sample volume at 10 T. Therefore, the transition to the CDS does not occur simultaneously in the whole sample. This effect will be much bigger in a resistive coil where the homogeneity is 20 times worse. However, we will see below, that the third harmonic rectification does not affect the determination of the critical field of the CDS. In Fig. 4(a) the phase angle of the third harmonic is shown for the same conditions like in Fig. 3. For magnetic fields where the amplitude of the third harmonic is below the noise level its phase angle is not determined. Therefore, the phase varies between -180 to +180 degree. With the appearance of a third harmonic signal at the transition to the CDS the phase becomes finite. This passage has a threshold character as well. The arrow in Fig. 4(a) shows the position of the threshold which yields the critical field µ 0 H c3 = 9.7 T.

The behavior of the out-of-phase part of the first harmonic response, shown in Fig. 4(b), offers another possibility to determine the critical field. In the uniform state, without domains, the imaginary part is small and varies smoothly especially at low magnetic field due to the magnetoresistance and changing eddy currents. After the transition to the CDS the out-of-phase signal changes rapidly. The transition point can be found as the intersection of two lines, as it is shown in Fig. 4(b). Here, we obtain for the critical field µ 0 H c4 = 9.4 T.

A comparison of the values H c1...c4 for the transition field at 2.7 K shows that they are very close. We note that the critical field of about 10 T agrees roughly with the phase boundary found in the Hall probe experiments 8 . All above presented methods could, in principle, be used to determine the phase boundary of the CDS. The first method (fig. 3(a)) requires at least two field sweeps. Measurements of the out-of-phase part (fig. 4(b)) are not precise, due to the high conductivity of silver and the resulting eddy currents (The situation might be different in a less conducting metal). Therefore, for silver the third harmonic measurements to determine the CDS phase diagram are preferred. Moreover, we will see below that the obtained values with the third harmonic for the phase boundary (H, T ) do not depend drastically on the frequency and amplitude of the field modulation which offers the possibility to measure in noisier conditions. The found scattering in the values of the transition fields obtained from the different methods gives an uncertainty of about ±0.5 T in the transition fields.

IV. PHASE DIAGRAM

Because of the increased noise level in the water-cooled resistive magnets, we needed to increase the signal-tonoise ratio by using rather high modulation frequencies ≈160 Hz and higher modulation amplitudes, 1.0 G and more. In the following we check whether the modulation frequency and amplitude can be varied without changing the value of the critical field deduced from the third harmonic response.

Modulation amplitudes of the order of the width of the hysteresis loop are required to resolve the amplitude reduction in the normalized pickup voltage in Fig. 3(a). For the third harmonic signal, as shown in Fig. 2, the nonlinear features persist up to high modulation amplitudes. Figure 5 shows traces with 0.2 and 1.0 G modulation amplitude. For 1.0 G modulation amplitude there is a very small third harmonic signal before the transition to the CDS takes place. This small contribution to the third harmonic is due to the nonlinearity of the dHvA effect itself [START_REF] Shoenberg | Magnetic oscillations in metals[END_REF] . Nevertheless, the position of the sharp increase remains unchanged.

Figure 6 shows that increasing the modulation amplitude up to 10 G and varying the modulation frequency by a factor four between 40 Hz and 160 Hz does not change the position of the critical field, either. The results presented here were obtained in the resistive magnet. The measurements were made at low temperatures in order to compare them with data obtained in the superconducting magnet.

All results for the CDS transition points obtained in the superconducting and the resistive magnets are presented in Fig. 7. The critical fields for each temperature are found as the field where the third harmonic response starts to arise like in Fig. 6. One should note that near the flat maximum of the phase diagram T -sweep measurements would be in principle better. The solid line in Fig. 7 is the CDS boundary calculated for silver using the LK-formula for the susceptibility criterion χ = 1 with a Dingle temperature of T D = 0.2 K for our sample 13,18 . For comparison, theoretical diagrams for T D = 0.1 K and 0.8 K are shown in Fig. 7.

A good agreement of the phase diagram predictions based on the LK-formula with our data can be seen for a Dingle temperature T D = 0.2 K. Data points obtained in superconducting and resistive magnets overlap which supports that the different measurement conditions did not affect the precise determination of the phase boundary.

V. CONCLUSION

Like previously observed in beryllium 11 , we have shown that hysteresis appears in silver in the Condon domain state. This substantiates that hysteresis is likely to occur in all pure metals that exhibit Condon domains. The hysteresis leads to with threshold character arising extremely nonlinear response to a modulation field. In particular, the third harmonic response to a modulation field increases sharply upon entering into the Condon domain state. This offered the possibility to determine easily the CDS phase boundary with high accuracy.

The critical fields obtained from the third harmonic of the pickup signal of the ac modulation technique, turned out to be independent on changes of the modulation frequency and amplitude. Due to this independence this method could be used with higher modulation frequencies in pulse magnetic fields.

Very good agreement of the CDS phase diagram is found with calculations of the dHvA signal based on the LK-theory. This agreement shows that the LK-formula describes well the field dependent magnetization of the nearly spherical Fermi surface of silver. Furthermore, the agreement demonstrates that the described method is correct for the determination of the CDS phase diagram.

FIG. 3 :

 3 FIG. 3: (Color online) (a) Pickup voltage normalized on the modulation level for low and high modulation level. Up to about 10 T the response is linear with respect to the modulation level. (b) Third harmonic of the pickup voltage measured at 0.2 G modulation amplitude showing that starting from 9.5 T the harmonic content in the response increases steeply. Lower part of the figure shows respective zooms. Both data measured at 2.7 K.

FIG. 4 :

 4 FIG. 4: (Color online) (a) Phase angle of the third harmonic showing clearly the transition between noise outside the CDS and a fixed phase in the CDS. (b) The out-of-phase part of the first harmonic response changes due to the arising hysteresis. Both data measured at 2.7 K and 0.2 G modulation level.

FIG. 5 :

 5 FIG.5:(Color online) Third harmonic response for two modulation levels measured in a superconducting coil at a modulation frequency of 160 Hz at 2.7 K. The same critical field is found from the steep increase.

FIG. 6 :

 6 FIG.6: Third harmonic response measured after averaging over the oscillations in a resistive coil for different modulation amplitudes (a) and modulation frequencies (b).The critical field values obtained at a given temperature are independent on modulation frequency and amplitude.

FIG. 7 :

 7 FIG.7:(Color online) Phase diagram in the (H, T ) plane for silver. Experimental points from the superconducting and resistive magnet. The solid line is the CDS boundary calculated by the LK-formula for TD = 0.2 K13,18 . The dashed and the dotted lines correspond to T D = 0.8 K and TD = 0.1 K, respectively.

Acknowledgments

We are grateful to I. Sheikin and V. P. Mineev for fruitful discussions.