**1. Introduction**

Cadmium (Cd) and lead (Pb) are metals that have no biologic role in humans [1–4]. All of their perceptible e ffects are toxic [1–4]. Indeed, Cd and Pb are two of ten chemicals listed by the World Health Organization (WHO) as environmental pollutants of major public health concern [5]. Tissues and organs accumulate Cd and Pb because no excretory mechanism has evolved to eliminate these metals [6–8]. Consequently, tissue levels of Cd and Pb increase with age, as do risks of common ailments that are often viewed as outcomes of aging. Although the highest concentrations of Cd and Pb are found, respectively, in kidneys and bone, toxic e ffects of these metals are not confined to diseases

of the kidney and skeleton [1–4,9,10]. It has been estimated that dietary intake of Cd, Pb, inorganic arsenic, and methylmercury have resulted in 56,000 deaths and more than 9 million disability-adjusted life-years worldwide [11]. For the nonsmoking population of adults, diet is the main exposure source of Cd and Pb [2,12–16].

Oxidative stress and inflammation have been identified as common toxic mechanisms of Cd and Pb even though neither metal undergoes a change in valence (redox inert) [17–22]. Both are primarily divalent [22–24]. In addition, Cd has a similar ionic radius to that of calcium (Ca) and electronegativity similar to that of zinc (Zn), and both Cd and Pb exhibit higher a ffinity than Zn for sulphur-containing ligands (Cd > Pb > Zn) [23–26]. Consequently, displacement of Zn and Ca and disruption of Zn and Cu homeostasis are other plausible toxic mechanisms [27–35]. All sulphur-containing amino acids, peptides and proteins with functional thiol (-SH) groups are potential ligands (molecular targets) for Cd and Pb. Examples include glutathione (GSH), numerous enzymes, zinc-finger transcription factors, and the metal-binding protein metallothionein (MT) [23,24,36]. Through Zn displacement, Pb impairs the activity of delta-aminolevulinic acid dehydratase (δ-ALAD), an enzyme required for the biosynthesis of heme, which is the functional group of hemoglobin, nitric oxide synthase, and cytochromes of the mitochondrial respiratory chain and xenobiotic metabolism [37]. Inhibition of the calcium-permeable acid-sensing ion channel may be the mechanism that accounts for the neurotoxicity of Pb [38,39].

The goal of this review is to provide an update on health risks associated with the low-to-moderate levels of environmental Cd and Pb to which most populations are exposed. High-dose exposure, which is relatively rare, is outside the scope of this review. We sought to establish dose–response relationships between ingested amounts of these toxic metals and parameters of tubular cell injury that have been associated with loss of glomerular filtration. This information is relevant to public health policy regarding advisable exposure limits. We discuss the interim safe intake level for Pb, the current tolerable dietary intake level for Cd, and the concept of the threshold level of urinary Cd. We describe the main dietary sources of Cd and Pb and the estimated intake levels of these metals among average and high consumers of tainted food. We highlight the utility of blood and urinary levels of Cd and Pb as indicators of internal accumulation of the metals. A connection is elaborated between Cd-induced tubulopathy and a decrease in glomerular filtration rate (GFR) to levels commensurate with chronic kidney disease (CKD). Epidemiologic data linking Cd and Pb exposure to enhanced CKD risk are summarized, as are data from longitudinal studies showing that Cd and Pb exposure may increase mortality from cancer and cardiovascular disease.

#### **2. Health Risk Assessment of Chronic Exposure to Cadmium and Lead**

#### *2.1. The Critical Target of Toxicity*

Long-term chronic exposure to Cd and Pb has been associated with distinct pathologies in nearly every tissue and organ throughout the body [1–4,14,25]. However, in health risk assessment, the kidney was considered to be the critical target of Cd toxicity [1,8], while the brain was the critical target of Pb toxicity [3,4,25]. Accordingly, dietary intake estimates associated with a significant increase in the risk of nephrotoxicity of Cd or neurotoxicity of Pb were used to derive a tolerable intake level. One method to evaluate whether a given food contaminant poses a health risk is to compare dietary intake estimated by total diet studies with the provisional tolerable weekly intake (PTWI), as established by the Joint Expert Committee on Food Additives and Contaminants (JECFA) of the Food and Agriculture Organization (FAO) and the WHO of the United Nations (FAO/WHO).

#### *2.2. Tolerable Intake Levels*

The PTWI for a chemical was defined as an estimate of the amount of a given chemical that can be ingested weekly over a lifetime without an appreciable health risk. The PTWI figures were first provided for Cd and Pb in 1989 and then amended in 1993 and 2010 [40,41]. The 1993 PTWI figures

for Cd and Pb were 7 and 25 μg per kg body weight per week, respectively. In 2010, the PTWI for Cd was amended to a tolerable monthly intake (TMI) level of 25 μg per kg body weight per month. This intake level is equivalent to 0.83 μg per kg body weight per day or 58 μg per day for a 70-kg person [41]. The model for deriving PTWI and TMI of Cd was based on elevated β2-microglobulin (β2MG) excretion as the sole evidence of nephrotoxicity [41]. In Section 3.1, we provide current Cd intake levels in various countries and their sources.

For Pb, the previously established PTWI of 25 μg per kg body weight per week was withdrawn because it did not a fford health protection [41]. A new tolerable Pb intake level could not be established as dose–response analyses indicated that no threshold levels exist for neurotoxicity of Pb. Thus, no amount of Pb intake is safe, and no tolerable Pb intake level has been o fficially identified. However, the U.S. Food and Drug Administration (FDA) has proposed a dietary Pb intake level of 12.5 μg/day as an interim safe intake level for the general population of adults [42,43]. This intake level corresponds to a blood concentration of Pb ([Pb]b) of 0.5 μg/dL, which has not been found to be associated with an adverse e ffect in adults in any epidemiologic studies. In Section 3.1, we provide current Pb intake levels in various countries and their sources.

#### *2.3. Urinary Cd Threshold Level*

A urinary Cd excretion rate (ECd) of 5.24 μg/g creatinine was adopted as a threshold limit [41]. However, the established threshold level is questionable. Chronic environmental exposure to low-level Cd, producing urinary Cd one-tenth of the conventional threshold, has been associated with deterioration of kidney function, as assessed with estimated GFR (eGFR) [44–46]. A urinary Cd concentration ([Cd]u) as low as 1 μg/L, corresponding to blood Cd concentration ([Cd]b) of 0.5 μg/L, was associated with an increased risk of eGFR less than 60 mL/min/1.73 m2 [44,47]. It can be argued that risk of nephrotoxicity of any toxicants, Cd and Pb included, should be based on eGFR, which is a reliable measure of kidney function and diagnosis and staging of CKD [48–50]. A dose–response analysis of urinary Cd and eGFR, rather than of urinary Cd and β2MG, indicates that Cd-induced nephrotoxicity occurs at a much lower ECd than previously thought [12,51–54]. We believe that the established TMI for Cd is not protective of kidneys, just as the 1993 PTWI for Pb does not prevent neurotoxicity. The 1993 PTWI for Pb has now been withdrawn [41]. In Section 4.3.4. we provide an in-depth analysis of β2MG excretion in Cd nephropathy.

#### **3. Exposure Sources and Dietary Intake Estimates**

For the general nonsmoking population of adults, the diet is the major exposure source of both Cd and Pb. In this section, both natural and anthropogenic sources of Cd and Pb in the human diet are highlighted. In addition, a reliable dietary assessment and food safety monitoring method, such as a total diet study, is discussed, and estimated intake levels of Cd and Pb derived from recent total diet studies in various countries are provided.

#### *3.1. Environmental Sources of Cadmium and Lead*

Volcanic emissions, fossil fuel and biomass combustion, and cigarette smoke are sources of Cd and Pb released as CdO and PbO [55–59]. Experimental studies have shown that inhaled CdO and PbO are more bioavailable than oral Cd and Pb [60–63]. Typically, potable water is not a source of Cd or Pb, except in cases where significant amounts of Pb plumbing have been used, as occurred in the recent Flint, Michigan, water crisis [64,65].

Years of production and industrial use of Cd and Pb have mobilized these metals from nonbioavailable geologic matrices to biologically accessible sources from which they can enter food chains [55]. Like all other metals, Cd and Pb are not biodegradable and thus can persist indefinitely in the environment [55]. The use of contaminated phosphate fertilizers has also added these toxic metals to agricultural soils [6,7], causing a further increase in Cd and Pb in the food chain [66–68]. Livestock that graze on contaminated pastures can accumulate Cd in the kidney and liver at levels that make

these organs unsafe for human consumption [69]. In Pb-exposed cattle, blood Pb levels correlated with levels of Pb in liver, bone and kidney, but not in brain or skeletal muscle (beef) [70]. Of note, a detectable amount of Pb was found in beef at a blood Pb concentration of 4.57 μg/dL. This blood Pb level was close to the exposure limit for neurotoxicity of Pb in children (5 μg/dL) [71]. Molluscs and crustaceans accumulate Cd and are also notorious hyperaccumulators of other metals [72–75]. For most species, fish muscle does not appear to be a significant source of Cd and Pb, but there are exceptions, as indicated in Section 3.1 [76].

In a similar manner to molluscs and crustaceans, plants have the propensity to concentrate Cd and Pb from the soil. Plants have evolved multiple metal detoxification mechanisms, including an array of metal-binding ligands such as MT, phytochelatins (PCs), other low-molecular-weight thiols, GSH, cysteine, γ-glutamylcysteine, and cysteinylglycine [77–79]. As Cd exerts toxicity in the "free" ion or unbound state, complexes of Cd and metal-binding ligands, such as CdMT and CdPC, are viewed as detoxified forms [80]. Accordingly, the various types of metal-binding ligands render plants capable of tolerating levels of Cd and Pb that are toxic to animals and humans.

Owing to their phylogenic characteristics, tobacco, rice, other cereal grains, potatoes, salad vegetables, spinach, and Romaine lettuce accumulate Cd more efficiently than other plants [81]. An outbreak of "itai-itai" disease, a severe form of Cd poisoning from contaminated rice, serves as a reminder of the health threat from Cd contamination of a staple food crop [82].

#### *3.2. Total Diet Studies and Dietary Intake Estimates*

Reliable methodology is vitally important to assess the levels of contaminants in commonly eaten foods and to set food safety standards. The total diet study has been widely used by authorities to estimate intake levels and identify sources of Cd and Pb in the human diet [83–87]. It is also known as the "market basket survey" because samples of foodstuffs are collected from supermarkets and retail stores to determine levels of nutrients, food additives, pesticide residues and contaminants [2,83–87]. It serves as a food safety monitoring program that provides a basis to define a maximally permissible concentration of a given contaminant in a specific food group.

In a typical total diet study, an intake level of a given contaminant from a study food item (rice as an example) is computed based on an amount of the food item consumed per day and the concentration of a contaminant in the rice samples that are analyzed in a study. The median and 90th percentile concentration levels of a contaminant are used to represent the intake levels of a contaminant by average and high consumers, respectively [88].

Table 1 summarizes most recent total diet studies showing intake levels of Cd among adult consumers in China [89–91], Korea [92,93], Germany [94], Spain [95,96] and the U.S. [97–99] along with the list of foods that contributed significantly to total intake of the metal. Table 1 summarizes also food products that contributed significantly to total intake of Pb and the estimated intake levels of the metal among adult consumers in China [89–91], Korea [92], Germany [100], Spain [95] and the U.S. [84]. Furthermore, Cd intake levels estimated for consumers in Sweden [88] France [101], Belgium [102] and a region with Cd pollution of Japan [103] are provided.

**Table 1.** Estimated intake levels of cadmium and lead and their sources.

A current tolerable Cd intake level established by FAO/WHO for the population of adults is 25 μg per kg body weight per month (58 μg per day for a 70-kg person) [41]. No tolerable Pb intake level has been identified after a previously established guideline was withdrawn in 2010 [41]. U.S. FDA interim safe intake level of Pb for the population of adults is 12.5 μg per day [42].

#### 3.2.1. Estimated Cadmium Intake Levels in Various Populations

In a recent total diet study in China, the Cd intake among average consumers was 32.7 μg/day, with rice and vegetables being the main sources [89]. In Mongolia, potatoes were the main source of Cd, contributing 24% of the total Cd intake [89]. Nori, peanuts, squid, cuttlefish and mushrooms had relatively high Cd contents [90,91].

The Cd intake level among average consumers in South Korea was 12.6 μg/day [92]. Cereals and vegetables, beverages, fruits and nuts, and dairy products (milk included) were the main dietary sources. Cereals, oily seeds and fruits, and vegetables had relatively high Cd concentrations. A higher average Cd intake of 22 μg/day was reported in another Korean study (*n* = 1245), where Cd intake in gastric cancer cases was compared to noncancer controls [93]. An average amount of rice consumed by the control group was 587.3 g/day versus 610.9 g/day in gastric cancer cases. Rice was the major contributor (40.3%) to total Cd intake, followed by squid (11.8%), eel (11.0%), crab (8.6%), shellfish (3.6%), kimchi (Korean cabbage; 3.5%) and seaweed (3.5%).

The Cd intake levels among average and high consumers in Germany were 14.6 and 23.5 μg/day, respectively [94]. Cereals and vegetables were the main Cd sources, followed by beverages, fruits and nuts, and dairy products (milk included). Cereals, oily seeds and fruits, and vegetables had relatively high Cd contents [94].

The Cd intake among average consumers in Spain was 7.7 μg/day, with cereals and fish as the main sources, contributing to 38% and 29% of total Cd intake, respectively [95]. In another dietary study of 281 postmenopausal women in Spain, an average Cd intake was 30 (range, 20−41) μg/day [96]. These data illustrate that when the total diet study methodology is used, Cd intake from the diet showed a little variation and, in many cases, probably represented an underestimation of actual Cd intake.

The Cd intake among average consumers in the U.S. was 4.63 μg/day. Cereals and bread, leafy vegetables, potatoes, legumes and nuts, stem/root vegetables, and fruits contributed, respectively, to 34%, 20%, 11%, 7% and 6% of total intake [97]. Spaghetti, bread, potatoes and potato chips were the top three Cd sources, followed by lettuce, spinach, tomatoes, and beer. Lettuce was an important Cd source for whites and blacks. Tortillas and rice were the main Cd sources for Hispanic Americans, Asians and some other ethnicities. However, a higher dietary Cd intake of 10.9 μg/day was recorded in the U.S. Women's Health Initiative study [98]. The average Cd intake of 4.63 μg/day by adults in the U.S. was close to a median Cd intake of 5 μg/day reported for consumers in Northern Italy, where cereals, vegetables and sweets were the main Cd sources [99].

It is noteworthy that the average Cd intake in Sweden (10.6 μg/day), France (11.2 μg/day) and Belgium (9.8 μg/day) was higher in all of these countries than the average Cd intake in the U.S. of 4.63 μg/day [88,101,102]. For average consumers in France, bread and potato-based products contributed, respectively, to 35% and 26% of total Cd intake, while potatoes and wheat combined contributed to 40−50% of Cd intake among average consumers in Sweden. Likewise, cereal products and potatoes were the main sources, contributing more than 60% to total Cd intake among average consumers in Belgium [102]. For high consumers in Sweden, average Cd intake was 23 μg/day, with seafood (shellfish) and spinach being additional Cd sources [88]. For high consumers in France, Cd intake was 18.9 μg/day, with additional Cd coming from molluscs and crustaceans [101]. Cd contents in molluscs and crustaceans, o ffal, sweet and savoury biscuits and cereal bars, and chocolate in a French total diet study were 0.167, 0.053, 0.030, and 0.029 mg/kg, respectively.

In summary, the average Cd intake levels in China (32.7 μg/day), Korea (12.6 μg/day), Germany (14.6 μg/day), Spain (7.7 μg per day) and Sweden, France and Belgium (range, 9.81−11.2 μg/day) were all higher than the average Cd intake in the U.S (4.63 μg/day) [88,101,102]. For average consumers, staple foods were the main Cd sources. Seafood (shellfish), o ffal, spinach, lettuce and chocolate were additional sources of Cd among high consumers. These estimated Cd intake levels did not exceed a current FAO/WHO tolerable intake level of 58 μg/day [41]. However, evidence for adverse health e ffects has emerged from cross-sectional and longitudinal studies of populations from these countries, in which urinary and/or blood Cd levels were used to quantitate exposure levels (Section 5). Thus, the utility of dietary intake estimates in health risk assessment is questionable. In another example, the total diet study undertaken in two areas of Japan affected by Cd pollution reported the median Cd intake levels in female farmers residing in the two areas as 47.8 and 55.7 μg/day [103]. These estimated Cd intake levels in Cd-polluted areas were lower than the FAO/WHO tolerable intake level, but adverse health effects were observed [103].

#### 3.2.2. Estimated Lead Intake Levels in Various Populations

The Pb intake level among average consumers in China was 35.1 μg/day, with cereals, meats, vegetables, beverages and water as the main sources [89]. These food and beverage items together contributed to 73.26% of total Pb intake [89]. Kelp, nori, processed and preserved soybean, meat, and fungus products had relatively high Pb concentrations [90]. For the population of Jiangsu province, a higher mean Pb intake level of 73.9 μg/day was reported, with cereals and vegetables as the main sources, contributing to 57% of total Pb intake [91].

The Pb intake level among average consumers in South Korea was 9.8 μg/day [92]. High levels of Pb were found in a range of products, notably seaweed, shellfish and crustaceans, molluscs, fish, sugar and sugar products, and beverages (fruit juice, carbonated fruit juice, carbonated drinks, sports drinks and coffee).

The Pb intake levels among average and high consumers in Germany were 37.1 and 50.4 μg/day, respectively [100]. Beverages, vegetables, fruits and nuts and cereals were the main Pb sources. Foods with relatively high Pb concentrations were meat (offal included), fish (seafood), vegetables and cereals [100].

The Pb intake level among average consumers in Spain was 14.7 μg/day, with cereals as the main source at nearly half (49%) of total intake [95]. Sweeteners and condiments, vegetable oils, meat, and fish had relatively high Pb concentrations [95].

The Pb intake levels among average and high consumers in the U.S. were 1.7−5.3 and 3.2−7.8 μg/day, respectively [84]. Grains, beverages, vegetables, dairy products, fruits, meat, and poultry plus fish contributed to 24.1%, 14.3%, 10.7%, 9.7%, 9.3% and 3.4% of total intake, respectively. Foods with relatively high Pb concentrations were chocolate syrup, liver, canned sweet potatoes, brownies, low-calorie buttermilk, salad dressing, raisins, English muffins, canned apricots, milk chocolate, candy bars, chocolate cake, chocolate chip cookies, wine and oat ring cereal.

In summary, average Pb intake in China (35.1 μg/day) was close to the level of intake in Germany (37.1 μg/day). These Pb intake levels were higher than the intake figures estimated for average consumers in Spain (14.7 μg/day), Korea (9.8 μg/day) and the U.S. (1.7−5.3 μg/day). Pb intake from the diet was highest (50.4 μg/day) among high consumers in Germany. To date, a tolerable Pb intake level has not been identified after a tolerable intake level of 25 μg per kg body weight per week was withdrawn in 2010 [41]. An interim tolerable intake level for Pb of 12.5 μg per day has been proposed for the population of adults by the U.S. FDA [42,43]. Based on this interim safe-intake figure (12.5 μg/day), dietary Pb intake levels among average consumers in China, Germany and Spain could be considered as excessive and may pose a significant health risk.

#### *3.3. Absorption of Cadmium and Lead: An Overview*

As the body does not synthesize nor break down metals, transporter systems and pathways have evolved to acquire all required elements, notably calcium (Ca), zinc (Zn), manganese (Mn), copper (Cu), and iron (Fe), from exogenous sources [104]. These metal transporters and pathways also serve as routes of entry for toxic metals Cd and Pb in the diet [105–110]. Early work suggested that the iron (Fe2+) transporter, divalent metal transporter1 (DMT1), was a likely route of entry for Cd and Pb [111,112]. Later, it was demonstrated that Pb entered enterocytes through a mechanism that was independent of DMT1 [113,114]. Additionally, although DMT1 has the same high affinity for Cd2+ as it does for Fe2+ [112], ferroportin 1 (FPN1) exports iron but not Cd [115]. Calbindin-D28k, a calcium-binding protein, may transport Cd to the basolateral membrane of enterocytes and may export Cd into the portal blood circulation [110,116].

Receptor-mediated endocytosis and transcytosis are the likely mechanisms for absorption of the Cd–metallothionein complex (CdMT) and the Cd–phytochelatin complex (CdPC) [117–119]. The specific metal transporters, carriers and receptors that have been implicated in the absorption of Cd and/or Pb include DMT1, a Zrt- and Irt-related protein (ZIP) of the zinc transporter family, the Ca2+-selective channel TRPV6, and the human neutrophil gelatinase-associated lipocalin (hNGAL) receptor [110,120]. ZIP14 and TRPV6 are highly expressed by intestinal enterocytes [107–109,120].

#### *3.4. The Kinetics of Cadmium and Lead in the Human Body*

Figure 1 outlines exposure sources of Cd and Pb, their entry routes, tissue distribution, storage organs and targets of toxicity.

**Figure 1.** Entry routes, distribution, storage and urinary excretion of cadmium and lead.

Absorbed Cd and Pb are transported to the liver. Inhaled CdO and PbO are transported to the lungs. Cd induces the synthesis of MT in both liver and lung, and CdMT is formed. Hepatic and pulmonary CdMT are later released and contribute to [CdMT]b. Pb does not induce MT in the liver or lung, and it binds presumably to GSH and other thiols in these organs. The fractions of absorbed Cd and Pb that are not taken up by the liver in the first pass then reach systemic circulation and are taken up by tissues and organs throughout the body, kidneys included. Pb is taken up mostly by bone and is later released, contributing to [Pb]b. CdMT from all sources (liver, lung, intestine) passes glomerular membrane filtration, and the filtered CdMT is reabsorbed by proximal tubular cells. Small fractions of Cd and Pb in the body are excreted in urine. [Cd]u provides a useful indication of kidney injury and kidney burden. [Pb]u serves as a proxy for [Pb]p.

Oxides of Cd and Pb in the air reach the lung, prompting an increase in the synthesis of MT, and CdMT is formed in situ [121]. Presumably, the induction of MT synthesis in the lung is attributable to Cd only because Pb is a weak inducer [122]. The pulmonary CdMT is later released into systemic circulation and redistributed to the kidneys [123,124]. Of note, while inhaled Cd increased pulmonary MT synthesis, a study in mice has shown that low-dose oral Cd did not induce pulmonary MT [125].

Cd and Pb in the diet are transported via the hepatic portal system to the liver, where Cd induces copiously the synthesis of MT, to which it becomes tightly bound as CdMT [126]. Pb does not induce MT synthesis in the liver. An increase in hepatic MT protein levels was a result of elevated plasma levels of interleukin 6 that followed Pb administration [127]. Of note, hepatic MT synthesis is intensified in rats given Pb plus Cd, although oral Pb alone does not a ffect the synthesis of MT in the liver [128]. Presumably, in hepatocytes, Pb initially forms complexes with GSH [24]. A small fraction of hepatic Cd and Pb is excreted in faeces via bile or may be reabsorbed and returned to the liver [129]. Hepatic CdMT is also continuously released into systemic circulation. This hepatic process contributes to [Cd]b and the redistribution of Cd (as CdMT) to kidneys long after exposure cessation. In this manner, the liver serves as a reservoir of Cd. The fraction of dietary Cd and Pb not taken up by hepatocytes in the first pass reaches systemic circulation and is taken up by tissues and organs throughout the body. Pb is preferentially taken up by bone, where it is stored and later released [130]. Thus, bone serves as a reservoir of Pb, and this organ contributes to [Pb]b, as is discussed further in Section 4.2.

Through systemic circulation, CdMT and CdPC of dietary origin and intestinal and hepatic CdMT [126] all reach the kidneys, where they are filtered by glomeruli by virtue of their small molecular weights [8]. Once they pass the glomerular membrane into the tubular lumen, they are then subsequently reabsorbed by proximal tubular epithelial cells as these cells are equipped with protein internalization mechanisms [131–134]. Due to a lack of biologically active mechanisms to eliminate excess metals, nearly all reabsorbed Cd is retained in the kidneys.

In summary, ingested Cd is mostly reabsorbed and sequestered in kidneys [126,131–134], while ingested Pb is taken up and retained in bone [135]. Only a small fraction of Cd and Pb in the kidneys is excreted in urine (Section 3.4, Figure 2). An in-depth discussion on the source of excreted Cd is provided in Section 4.3.2. In e ffect, the kidney Cd content reflects cumulative lifetime exposure, while bone Pb is an indicator of Pb body burden.

#### *3.5. Cadmium and Lead Accumulation in Kidneys and Urinary Excretion: Australian Experience*

Figure 2 provides data on Cd and Pb levels in kidney cortex samples ([Cd]k, [Pb]k) from a subgroup of an Australian autopsy study (*n* = 42) [136,137]. The mean age, body mass index (BMI) and kidney weight were 38 years, 23.5 kg/m2 and 267 g, respectively. Kidney weight relative to body weight showed a decrease in old age (Figure 2A). The means of [Cd]k and [Pb]k were 14.5 and 4.16 μg/g kidney cortex wet weight, respectively. Thus, the kidney cortex accumulated 3.5 times greater amounts of Cd than Pb on a weight basis and 6.4 times higher amounts of Cd than Pb on a molar basis (Figure 2B). In an analysis of Cd and Pb concentrations in urine samples ([Cd]u, [Pb]u) from 17–23 urinary bladders of subjects in this subgroup, [Cd]u increased linearly with age and [Cd]k (Figure 2C,D). Distinctly, [Pb]u did not correlate with age or [Pb]k. The correlation between [Cd]u, age and [Cd]k supports the use of [Cd]u as a kidney burden indicator. The mean [Pb]k of 4.16 μg/g wet weight observed in this Australian study was much higher than the median [Pb]k of 0.08 μg/g wet weight found in Swedish kidney transplant donors (*n* = 109, 24–70 years, median 51), but there was only a small di fference in [Cd]k in Australian and Swedish studies (14.5 vs. 12.9 μg/g wet weight) [138].

**Figure 2.** Cadmium and lead accumulation levels in kidneys and their levels in urine.

Figure 2A is a scatterplot that relates % kidney weight to body weight ratios to age in a subgroup of an Australian autopsy study (*n* = 42, mean age 38 years). Figure 2B provides data on age, BMI, kidney weight and Cd and Pb contents in kidney cortex ([Cd]k, [Pb]k). Figure 2C,D are scatterplots showing significant correlations between [Cd]u versus age and [Cd]u versus [Cd]k, respectively. These correlations support the utility of [Cd]u in kidney burden assessment. Miniscule correlations between [Pb]u versus age and [Pb]u versus [Pb]k are indicated by scatter plots in Figure 2E,F, respectively.

#### **4. Dosimetry and Nephrotoxicity Assessment**

The dogma in health risk assessment states that for a given toxicant to pose a health risk, it must enter the body to reach its critical target of toxicity. The levels of a toxicant in body fluid and tissues and organs may serve as dosimetric matrices. In this section, we discuss the utility of urine, whole blood, blood plasma (serum), and erythrocytes in the assessment of Cd and Pb body burden. The manifestation of nephrotoxicities such as the leakage of intracellular proteins into the filtrate, a reduction in glomerular filtration rate, and impaired reabsorption of filtered proteins is discussed.

### *4.1. Cadmium Dosimetry*

#### 4.1.1. Blood Cadmium Versus Urinary Cadmium

In the studies examining decreases in [Cd]b and [Cd]u in groups of Cd-exposed Swedish workers after exposure had ceased, the estimated half-life values varied from 75 to 128 days for the fast component, representing [Cd]b [139,140]. For the slow component, representing Cd in tissues, the estimated half-life values varied from 7.4 to 16.0 years. A comparable half-life figure of 14 years (range, 9−28 years) was derived from a Japanese study [141]. In a separate analysis, the half-life of whole-body Cd in Japanese men with lower exposures ([Cd]u < 5 μg/L) was 23.4 years, nearly two-fold longer than the half-life figure of 12.4 years, estimated for those with higher exposures [142]. A whole-body half-life of Cd over 45 years was estimated based on data from Swedish kidney transplant donors, who had much lower [Cd]u, [Cd]b and [Cd]k than those in Japanese studies [143]. Collectively, these findings indicate that Cd is mostly retained in the body, particularly the kidneys, in low-level exposure situations, while there is a substantial loss of [Cd]k due to Cd-induced tubular injury in chronic high-exposure conditions [54].

The half-life of whole-body Cd falls in the same range as estimated half-life for Pb in bone (Section 4.2). Apparently, however, the half-life of blood Cd (between 75 and 128 days) is much longer than the 10-day half-life of blood Pb [144]. The estimated long half-life of [Cd]b could be due to continuing contribution to [Cd]b from tissues and organs, notably, the liver and lungs. To derive a half-life figure specific to exogenously derived Cd requires knowledge of contribution from the main endogenous sources, liver, lung and bone included, as depicted in Figure 1. Thus, at any given time, [Cd]b is considered to reflect recent exposure together with a continuing contribution from whole-body Cd, especially the liver and lungs.

Distinct from [Cd]b, urinary excretion of Cd (ECd) is an indicator of cumulative lifelong exposure. ECd varies directly with [Cd]k (see Figure 2) and the determinants of [Cd]k, such as age, gender, smoking, metalwork history and level of regional Cd pollution [8,136,145–148]. In a Swedish study, the highest Cd concentrations in copper smelter workers and the controls were seen in kidneys, followed by liver, lung and brain [148]. Of note, ECd showed stronger and more consistent associations with nephrotoxicity than [Cd]b. ECd is discussed further in Section 4.3.2.

In an autopsy study in Poland, ECd of 1.7 μg/g creatinine corresponded to [Cd]k of 50 μg/g [145]. In another Swedish study, kidney transplant donors (*n* = 109) had a mean [Cd]k of 12.9 μg/g, and the "urine-to-kidney Cd ratio" of 1:60. Accordingly, ECd of 0.42 μg/g creatinine corresponded to [Cd]k of 25 μg/g wet kidney weight [146]. In another study of kidney transplant donors, the means for ECd, [Cd]b, and [Cd]k in women were 0.34 μg/g creatinine, 0.54 μg/L and 17.1 μg/g kidney wet weight, respectively [147]. The corresponding figures in men were 0.23 μg/g creatinine, 0.46 μg/L and 12.5 μg/g, all of which were lower than in women [147].

#### 4.1.2. Cadmium in Erythrocytes Versus Blood Plasma (Serum)

The vast majority of [Cd]b appears to be concentrated in the cytosol of red blood cells. The chloride/bicarbonate anion exchanger ([Cl−/HCO3−], AE1, SLC4A1), is the major Cd transporter responsible for its uptake [149–151]. High- and low-a ffinity "iron" transport mechanisms have also been suggested to mediate Cd uptake [152]. Of interest, these transport mechanisms are also responsible for Pb and Zn uptake by erythrocytes [152–154]. Although Cd is concentrated in red blood cells, the toxicity of Cd in these cells has not been examined. One potential toxic e ffect is that Cd causes erythrocytes to undergo suicidal death, known as eryptosis, the mechanism by which such injured

cells are removed [155,156]. Enhanced eryptosis, seen in smokers, has been linked to oxidative stress and inflammation [157].

The Cd that remains in the plasma is bound to the amino acid histidine and proteins such as MT, pre-albumin, albumin, α2-macroglobulin, and immunoglobulins G and A [126,158,159]. A range of low molecular weight thiols, including GSH, cysteine, cysteinylglycine, homocysteine, and γ-glutamylcysteine, are also possible carriers of Cd [160–162]. The total concentrations of these low molecular weight thiols are in the low μM range (12–20 μM), while albumin thiol is more abundant (~0.6 mM). As Cd in plasma is readily exchangeable with other metals in target tissues, [Cd]p is more relevant than [Cd]b (Cd in erythrocytes) to toxic injury in those tissues. This may explain the conflicting results when different indicators of exposure were used. For instance, an association was not observed between breast cancer risk and [Cd]b [163], while [Cd]u has been consistently associated with a significant increase in risk for cancer [164].

Presently, plasma/serum Cd is rarely used in health risk assessment. The true distribution of Cd in whole blood and serum (plasma) remains to be fully characterized. The relationship between [Cd]b and plasma Cd at varying exposure levels is not known, nor is there a way to predict plasma Cd from whole blood/erythrocyte Cd data that are abundantly available. Nonetheless, the utility of the serum Cd concentration has been demonstrated in the studies linking Cd to an increased risk of respiratory disease in a representative U.S. population. Serum Cd levels ≥ 0.73 μg/L were associated with a 2.5-fold increase in the risk of obstructive lung disease among participants in NHANES 2007–2010 [165], while serum Cd levels in the highest quartile were associated with a 2.8-fold increase in the risk of recurrent wheeze and asthma in NHANES 2007–2012 participants aged 20–79 years [166].

### *4.2. Lead Dosimetry*

#### 4.2.1. Blood Lead Versus Bone Lead

Blood Pb ([Pb]b) constitutes less than 2% of the total body content of Pb. Ninety-nine percent of blood Pb is found in the cytosol of erythrocytes, while 1% or less is present in plasma [167–170]. However, [Pb]b is still the most frequently used marker of Pb exposure in epidemiologic studies because higher quantities are detectable by common analytical instruments compared with measurements of Pb in bone and urine. [Pb]b of 5 μg/dL is recommended as an actionable level or an excessive exposure level in infants and children, while [Pb]b of 20 μg/dL is adopted as an excessive exposure level in workplace settings [71].

At any given time, [Pb]b reflects recent intake plus the contribution from bone Pb (cumulative lifetime exposure). In children, the bone contribution is more than 90%, while the bone contribution to [Pb]b in adults ranges from 45% to 55% [171–173]. An estimated half-life of [Pb]b was 10 days (9.96 ± 3.92) in a study of Chinese children with Pb toxicity, in whom both bone Pb and [Pb]b were measured [144]. This estimated half-life of [Pb]b is much shorter than the previously estimated figures that ranged from 8–11 months to 2–3 years [174]. A half-life of as long as 2 years for [Pb]b was suggested in a study of the time required for [Pb]b to fall by 50% in Pb-exposed children undergoing chelation therapy [175]. The differences in [Pb]b half-life figures are due mostly to a correction for bone contribution in a recent study [144].

Bone Pb constitutes approximately 94% and 74% of the total body burden of this metal in adults and children, respectively [144,176–178]. The half-life of bone Pb varies from 10 to 20 years [179–182]. In a study of Pb accumulation in different types of bone from Swedish copper–lead smelter workers and controls, the highest Pb concentration was found in finger bone, followed in order by vertebrae, iliac crest and sternum [183]. Among soft tissues, Pb concentration was highest in the liver, followed by kidney, lung and brain [184]. Consistent with this study, an Australian study reported that liver contained Pb levels two times higher than kidney cortex: the mean hepatic and kidney cortex Pb levels were 0.19 and 0.09 μg/g wet tissue weight, respectively [137]. Bone Pb levels correlated with serum Pb levels in adults who were exposed to Pb in the workplace [185].

In environmental exposure scenarios, Pb in tibia showed stronger and more consistent associations with neurotoxicity than blood Pb. This supports bone Pb to be a better biomarker of cumulative dosimetry or the body burden than blood Pb [186–188]. In addition to the tibia, Pb in the patella was associated with declines in cognitive test scores in older persons [189]. In a prospective cohort study of men, chronic, low-level Pb exposure, assessed with bone Pb, has been linked to hypertension [190] and incident coronary heart disease [14]. The Western dietary patterns have also been linked to elevated bone and blood Pb levels in men [13,14].

#### 4.2.2. Plasma (Serum) Lead Versus Urinary Lead

For the same reason as plasma Cd (see Section 4.1.2), [Pb]p is more relevant than [Pb]b to toxicity in tissues/organs. The superiority of plasma Pb has been demonstrated in an early study of Pb-exposed Japanese workers, in which the strength of correlations between [Pb]p and urinary indicators of adverse effect (the inhibition of heme biosynthesis), namely, urinary coproporphyrin and urinary delta-aminolevulinic acid (ALA), was stronger than the correlations of [Pb]b with these indicators [191]. Although only 1% of whole-blood Pb is in the plasma fraction, this percentage increases sharply in high-exposure conditions, a phenomenon that suggests the binding capacity of the erythrocytes has been exceeded. A linear relationship between [Pb]p and [Pb]b has been demonstrated using data from Pb-exposed workers [182,192]. Likewise, a linear relationship between [Pb]p and [Pb]b was observed in a study of U.S. women of child-bearing age [183].

Notably, however, the measurement of [Pb]p can be problematic because hemolysis causes a spurious increase in [Pb]p. [Pb]u correlated more closely with [Pb]p than [Pb]b in a study of Pb-exposed workers in Japan [191]. [Pb]u may, thus, be used as a proxy for [Pb]p. The utility of urine in monitoring Pb exposure has been illustrated in studies in Japan [193], Belgium [194], China [195], Thailand [12] and the U.S. [196]. In a Chinese study, workers in a lead–zinc mine had higher [Pb]u, [Cd]u and urinary 8-hydroxydeoxyguanosine, an indicator of oxidative stress, compared with those who worked in a steel smelting plant [195].

In a Thai study, the 75th percentile level of [Pb]u was associated with a 2.3-fold increase in the risk of reduced eGFR [12]. This finding is reminiscent of studies relating ECd to eGFR (Section 4.3.2). In a follow-up study of 5316 participants in NHANES 1999–2010, [Pb]u levels > 1.26 μg/L were associated, respectively, with 1.79- and 6.60-fold increases in mortality from all causes and cancer [196].

#### *4.3. Assessment of Cadmium Nephrotoxicity*

As Figure 1 indicates, Cd enters the body from contaminated food, tobacco smoke, and polluted air. It is transported through the gut and lungs to the bloodstream, where it is bound primarily to albumin and taken up by red blood cells ([126], see Section 3.4). Most circulating Cd is eventually transferred to hepatocytes, which then synthesize the protein MT and store Cd in complexes of CdMT. As hepatocytes die, these complexes are released to the circulation, filtered by glomeruli, and reabsorbed primarily or entirely by proximal tubular cells [197,198]. After lysosomes within these cells degrade the complexes, Cd enters the cytoplasm, where it again induces the synthesis of MT, to which it is subsequently bound. Cd that remains free is believed to induce the release of copper (Cu), a transition metal, from MT; Cu then promotes the formation of reactive oxygen species, which inflict cellular injury [31,199]. The intensity of this injury is thought to be related to the concentration of free Cd in the cytoplasm [200,201]. As no mechanism exists for discharging Cd from intact tubular cells, the metal continues to accumulate in those cells as long as hepatocytes release CdMT to the bloodstream [202,203].

Commonly studied clinical expressions of tubular Cd toxicity include leakage of intracellular proteins into the filtrate, excretion of Cd itself, the estimated glomerular filtration rate (eGFR), and impaired reabsorption of small filtered proteins. Extreme toxicity also compromises reabsorption of filtered substances that are cotransported with sodium, including glucose, phosphate, urate, and amino acids. The resulting constellation of abnormalities, the so-called Fanconi syndrome, was once common in heavily contaminated regions of Japan [82]. As this syndrome is now a rare manifestation of Cd tubulopathy, it is not considered in this review.

#### 4.3.1. Release of Intracellular Proteins into the Filtrate

After cellular injury, proteins synthesized in tubular cells may be released into the filtrate and detected in urine. In studies of Cd nephropathy, N-acetyl-β-D-glucosaminidase (NAG) and kidney injury molecule 1 (KIM1) have been assayed most frequently.


Meet the editors

Professor Lăcrămioara Popa received her Ph.D. in Pharmacy in 2000. She is head of the Physical and Colloidal Chemistry Department, Faculty of Pharmacy, "Carol Davila" University of Medicine and Pharmacy (UMFCD), Bucharest, Romania, and president of the Ethics and Quality Assurance Commission at the same faculty. She is a member of the Council of Doctoral Studies of UMFCD and a Ph.D. supervisor (Habilitation

thesis, 2014). Professor Lăcrămioara graduated with several courses in Intellectual Property (WIPO Academy, Geneva) and was an expert evaluator at the Executive Unit for Financing of Higher Education, Romania. She has more than 28 years of experience in quality by design; analytical quality by design in the development, analysis, and optimization of pharmaceuticals; characterization of raw materials surface and pharmaceutical systems; and polymeric materials with biomedical applications.

Professor Mihaela Violeta Ghica received her Ph.D. in Pharmacy in 2008. She is a professor in the Physical and Colloidal Chemistry Department, Faculty of Pharmacy, "Carol Davila" University of Medicine and Pharmacy (UMFCD), Bucharest, Romania. She holds a master's degree in Physical Chemistry and Applied Radiochemistry from the University of Bucharest. She is a Ph.D. supervisor (Habilitation thesis, 2019). She has

more than 20 years of experience in modern methods of experimental statistical design in the development of drug delivery systems and technological processes optimization; biomaterials for tissue regeneration; obtaining, physico-chemical, biopharmaceutical, structural, and morphological characterization; and physical pharmacy of dermatocosmetics. She was a project director and partner responsible for several national and international projects

Professor Cristina Elena Dinu-Pîrvu received her Ph.D. in Pharmacy in 2004. She is a professor in the Department of Physical and Colloidal Chemistry, Faculty of Pharmacy, "Carol Davila" University of Medicine and Pharmacy (UMFCD), Bucharest, Romania. She holds a master's degree from the University of Agronomic Sciences and Veterinary Medicine, Bucharest, Romania. She is a Ph.D. supervisor (Habilitation

thesis, 2015), and director of the Research, Development and Innovation Department at UMFCD. She has more than 28 years of experience in the development of bio-/nanostructures for diagnosis and therapy; smart, efficient, safe biocompatible materials; and valorization of the biopharmacological potential of medicinal/ aromatic plants from spontaneous flora. She completed a postdoctoral fellowship at Polytechnic University, Bucharest, Romania.


**Abstract**

Hypoplastic left heart syndrome (HLHS) is a complex congenital heart disease (CHD) involving hypoplasia of the left ventricle (LV), aorta (Ao), and mitral valve. HLHS was uniformly fatal in the past, now survivable with 3-stage surgical palliation. However, there is high morbidity and mortality, with 25% of HLHS patients either dying or having a heart transplant within 1 year of age. The causes for such high morbidity and mortality are not well understood, but the majority of deaths are directly or indirectly related to cardiovascular/hemodynamics causes. Studies in a mouse model of HLHS uncover important contributing factors for single-ventricle patients such as the patient's intrinsic factors related to mitochondrial dysfunction, and derangements in the early stages of embryonic development. The HLHS mutant mice were noted to have metabolic dysfunction accompanied by cell cycle arrest and cardiomyocyte differentiation defects. Intrinsic cell defects may contribute to cardiac failure in the HLHS population. Moreover, strong evidence of the genetic etiology of HLHS has come from the observation that HLHS has a high recurrence risk and is associated with various chromosomal abnormalities. In this chapter, we will review the basic pathophysiology, pertinent pre-and post-operative managements of HLHS and recent advances derived from the HLHS mouse model.

**Keywords:** HLHS, cardiomyocyte, mitochondria, single ventricle, hemodynamics, complex genetics


**4.2.3 Deep N-H related acceptor state H2**

The ionized acceptor density (*N*A) is found to be in good linear dependence with N concentration in *p*-type GaAsN samples (see Fig. 12 (a)). As given in Figs. 12(b) and (c), the junction capacitance (*C*j) showed a *N*-related sigmoid behavior with temperature in the range 70 to 100 K. This behavior has not yet been observed in GaAs and *n*-type GaAsN grown by CBE. It was recorded at 20 K in silicon *p-n* junction and explained by the ionization of a shallow energy level (Katsuhata, 1978; 1983). Hence, the N dependence of *N*A and *C*j is explained by the thermal ionization of a *N*-related acceptor-like state. The thermal ionization energy of this energy level was estimated in the temperature range 70 to 100 K to be between 0.1 and 0.2 eV. It is in conformity with the theoretical calculations, which suggested the existence of a N-related hole trap acceptor-like defect with an activation energy within 0.03 and 0.18eV above the VBM of GaAsN (Janotti et al., 2003; Suzuki et al., 2008). Experimentally, a deep acceptor level, A2, was confirmed in CBE grown undoped GaAsN with ionization energies of *E*A1 = 130 ± 20 meV (Suzuki et al., 2008). On the other hand, the properties of *H*2 in N-varying GaAsN schottky junctions are cited below: The peak temperature of *H*2 is within the temperature range of increase of Cj. This means that the electric field at this temperature range fellows the same behavior of Cj and depends on N concentration. Hence, the emission of carriers from the charged traps is affect by the Poole-Frenkel emission (Johnston and Kurtz, 2006). This is confirmed by the fluctuation of *E*H2 from one sample to another depending on N concentration (see Table 3). However, the average of *E*H2 is within the energy range of the acceptor level obtained from theoretical prediction and identical to EA2 (Suzuki et al., 2008; Janotti et al., 2003). Furthermore, as given in Fig. 12 (d), *N*H2-adj is in linear dependence with *N* concentration. Therefore, *H*2 is proved to be the N-related hole trap acceptor-like state, which thermal ionization increased *C*j and

Investigation of Lattice Defects in GaAsN

H2 may be related to the N-H-VGa structure.

**0.1 0.2 0.3 0.4 0.5 0.6**

**[N](%)**

concentration at a growth temperature of 420 C.

undoped *p*-type GaAsN Schottky junctions.

Fig. 13. N dependence of (a) NA and (b) TEGa flow rate dependence of growth rate and N

[N](%) EH2 (eV) H2(cm2) NH2-adj(cm-3) Emax (V/cm) NH2-Cj(cm-3) NH2-Cj/ NA 0.12 0.210 2.8 10-14 2.64 1015 6.2 104 2.23 1015 0.36 0.20 0.150 6.3 10-16 3.08 1015 1.4 105 2.88 1016 0.91 0.34 0.138 6.3 10-16 5.20 1015 2.1 105 6.56 1016 0.93 0.51 0.103 1.3 10-17 9.12 1015 2.3 105 7.55 1016 0.87 Table 3. Summary of EH2, H2, *N*H2-adj, *E*max, *N*H2-est, and the ratio *N*H2-Cj/*N*A for CBE grown

**0.5 1.0 1.5 2.0 2.5 3.0**

**(b)**

**Growth rate (m/h)**

**0.00 0.05 0.10 0.15 0.0**

**TEGa(sccm)**

**0.4**

**0.6**

**0.8**

**[N] (%)**

**1.0**

**0**

**2**

 **N**

**A**

**(x 1016 cm-3)**

**4**

**6**

**8**

**(a)**

**10**

Grown by Chemical Beam Epitaxy Using Deep Level Transient Spectroscopy 507

To investigate the origin of *H*2, it is worth remembering some previous results about carrier concentration and the density of residual impurities in undoped GaAsN grown by CBE, obtained using Hall Effect, Fourier transform infra-red (FTIR), and second ion mass spectroscopy (SIMS) measurements. On one hand, under lower Ga flow rates (TEGa), *N*A and *N*H2 sowed a rapid saturation with [*N*], despite the increase of [*N*] (see Fig. 13 (a)). This means that the atomic structure of *H*2 depends on other atoms, either than *N*. In addition, the densities of *C* and *O* was found to be less than free hole concentration, which excludes these two atoms from the origin of *H*2. On the other hand, using SIMS measurements, the ratio [H]TEGa = 0.02/[H]TEGa = 0.1 was evaluated to be 0.6 (Sato et al., 2008). Furthermore, the free hole concentration at room temperature showed a linear increase with the density of *N-H* bonds (Nishimura et al., 2006). This means that *N*A depends strongly on [*H*] and the saturation of *N*A under lower TEGa can be explained by the desorption of *H* from the growth surface, since the growth rate in our films was found to be in linear dependence with TEGa. Hence, the structure of *H*2 is related to the N-H bond. However, the *N-H* bond may not be the exact structure of *H*2 because the slope of the linear relationship between *N*A and [*N-H*] increased with increasing growth temperature (TG [400, 430] C). This indicates that *N*A is determined by both the number of *N–H* and another unknown defect, which concentration increased with increasing TG. The binding energy of this unknown defect can be determined from Arrhenius plot. Furthermore, the formation energy of (N-H-VGa)-2 was found to be lower than (N-VGa)-3, (H-VGa)-2, and isolated VGa-3 (Janotti et al., 2003). This means that the unknown defect may be VGa. These predictions were experimentally supported using positron annihilation spectroscopy results (Toivonen et al., 2003). Hence,

drops the depletion region width. The contribution of *H*2 in the background doping of ptype GaAsN films grown by CBE can be evaluated from the ratio between the real NH2-Cj calculated from the change of Cj and *N*A at room temperature (Bouzazi et al., 2010). As shown in Fig. 12 (*e*) and (*f*), This ratio comes closer to the unit for a N concentration greater than 0.2%. Thus, *H*2 is the main cause of the high background doping in *p*-type GaAsN.

Fig. 12. N dependence of (a) *N*A, (c) amplitude of Cj after thermal ionization of *H*2, and (d) NH2-adj. (b) Sigmoid increase of Cj between 70 and 100 K for two different GaAsN samples. *N* dependence of (e) NH2-Cj and (f) NH2-Cj /NA.

drops the depletion region width. The contribution of *H*2 in the background doping of ptype GaAsN films grown by CBE can be evaluated from the ratio between the real NH2-Cj calculated from the change of Cj and *N*A at room temperature (Bouzazi et al., 2010). As shown in Fig. 12 (*e*) and (*f*), This ratio comes closer to the unit for a N concentration greater than 0.2%. Thus, *H*2 is the main cause of the high background doping in *p*-type GaAsN.

Fig. 12. N dependence of (a) *N*A, (c) amplitude of Cj after thermal ionization of *H*2, and (d) NH2-adj. (b) Sigmoid increase of Cj between 70 and 100 K for two different GaAsN samples. *N*

**2**

**0.4**

**0.6**

*N***H2-Cj/N**

**A** (**-**)

**0.8**

**1.0**

**(f)**

*N***H2-adj** **4**

**6**

(**x 1015cm-3**)

**8**

**(d)**

**10**

**0.0**

**0.7**

**Cjunction (x 100pF)**

**1.4**

**75 100 125 150**

**H2 [N] = 0.51% (b)**

**Tempertaure**

**0.1 0.2 0.3 0.4 0.5**

**[N] (%)**

**0.1 0.2 0.3 0.4 0.5**

**[N] (%)**

**[N] = 0.34%**

**0.0 0.2 0.4 0.6**

**[N](%)**

**0.0 0.2 0.4 0.6**

**[N] (%)**

**0.1 0.2 0.3 0.4 0.5**

**[N] (%)**

dependence of (e) NH2-Cj and (f) NH2-Cj /NA.

**0**

**0.0**

**0**

**2**

*N***H2-Cj**

(**x 1016cm-3**)

**4**

**6**

**8**

**(e)**

**10**

**0.3**

**Cj (x 100pF)**

**0.6**

**0.9**

**1.2**

**(c)**

**2**

 **N**

**( x 1016 cm-3)**

**A**

**4**

**6**

**8**

**(a)**

To investigate the origin of *H*2, it is worth remembering some previous results about carrier concentration and the density of residual impurities in undoped GaAsN grown by CBE, obtained using Hall Effect, Fourier transform infra-red (FTIR), and second ion mass spectroscopy (SIMS) measurements. On one hand, under lower Ga flow rates (TEGa), *N*A and *N*H2 sowed a rapid saturation with [*N*], despite the increase of [*N*] (see Fig. 13 (a)). This means that the atomic structure of *H*2 depends on other atoms, either than *N*. In addition, the densities of *C* and *O* was found to be less than free hole concentration, which excludes these two atoms from the origin of *H*2. On the other hand, using SIMS measurements, the ratio [H]TEGa = 0.02/[H]TEGa = 0.1 was evaluated to be 0.6 (Sato et al., 2008). Furthermore, the free hole concentration at room temperature showed a linear increase with the density of *N-H* bonds (Nishimura et al., 2006). This means that *N*A depends strongly on [*H*] and the saturation of *N*A under lower TEGa can be explained by the desorption of *H* from the growth surface, since the growth rate in our films was found to be in linear dependence with TEGa. Hence, the structure of *H*2 is related to the N-H bond. However, the *N-H* bond may not be the exact structure of *H*2 because the slope of the linear relationship between *N*A and [*N-H*] increased with increasing growth temperature (TG [400, 430] C). This indicates that *N*A is determined by both the number of *N–H* and another unknown defect, which concentration increased with increasing TG. The binding energy of this unknown defect can be determined from Arrhenius plot. Furthermore, the formation energy of (N-H-VGa)-2 was found to be lower than (N-VGa)-3, (H-VGa)-2, and isolated VGa-3 (Janotti et al., 2003). This means that the unknown defect may be VGa. These predictions were experimentally supported using positron annihilation spectroscopy results (Toivonen et al., 2003). Hence, H2 may be related to the N-H-VGa structure.

Fig. 13. N dependence of (a) NA and (b) TEGa flow rate dependence of growth rate and N concentration at a growth temperature of 420 C.

Table 3. Summary of EH2, H2, *N*H2-adj, *E*max, *N*H2-est, and the ratio *N*H2-Cj/*N*A for CBE grown undoped *p*-type GaAsN Schottky junctions.

Investigation of Lattice Defects in GaAsN

292-299.

401-408.

pp. 387-387.

pp. 6403-6406.

pp.<PHONE_NUMBER>04.

144.

2064.

881.

Grown by Chemical Beam Epitaxy Using Deep Level Transient Spectroscopy 509

Bouzazi, B.; Nishimura, K.; Suzuki H.; Kojima, N.; Ohshita, Y. & Yamaguchi M. (2010).

Bouzazi, B.; Suzuki, H.; Kojima, N.; Ohshita, Y. & Yamaguchi, M. (2011). Japanese Journal

Bube, R. H. (1956). Comparison of Surface-Excited and Volume-Excited Photoconduction in Cadmium Sulfide Crystals. Physical Review, Vol.101, pp. 1668-1676. Bube, R. H. (1960). Photoconductivity of Solids. John Wiley & Sons, Inc., New York, pp.

Deenapanray, P. N. K.; Tan, H. H. & Jagadish, C. (2000). Investigation of deep levels in

DeVore, H. B. (1959). Gains, response times, and trap distributions in powder

Friedman, D. J.; Geisz, J. F.; Kurtz, S. R.; & Olson, J. M. (1998). 1-eV Solar Cells with GaInNAs Active Layer. Journal of Crystal Growth, Vol.195, pp. 409-415. Geisz, J. F.; Friedman, D. J.; Olson, J. M.; Kurtz, S. & Keyes, B. M. (1998). Photocurrent of 1

Geisz, J. F. & Friedman, D. J. (2002). III–N–V Semiconductors for Solar Photovoltaic Applications. Semiconductor Science and Technology, Vol.17, No.8, pp. 769-777. Hall, R. N. (1952). Electron-Hole Recombination in Germanium. Physical Review, Vol. 87,

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Janotti, A.; Zhang, S. B.; Wei, S. H. & Van de Walle, C. G. (2002). Effects of hydrogen on

Janotti, A.; Wei, S. H.; Zhang, S. B. & Kurtz, S. (2003). Interactions between nitrogen,

Jock, M. R. (2009). Effect of N Interstitials on the Electronic Properties of GaAsN Alloy

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Katsuhata, M.; Koura, K. & Yoshida S. (1978). Temperature Dependence of Capacitance of

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Properties of Chemical Beam Epitaxy grown GaAs0.995N0.005 Homo-junction Solar


2.2.3. Construction Equipment

The research conducted by Dumitrescu and Delsenicu [28] identified that the equipment's complexity and maintenance adequacy can affect an operator's performance. Naskoudakis and Petroutsatou [31] emphasized the fact that, with regard to equipment development, new methods and designs are implemented to enhance reliability, machine control, comfort, and safety and to reduce the costs derived from failures and breakdowns, signifying the importance of the equipment's innovation as an influential factor. Their literature review also highlighted the importance of fleet management and construction equipment deployment. According to Vorster [4], fleet management should fulfill three overriding and critically important goals directly linked to human factors: (i) the equipment must be in the right place at the right time, (ii) the equipment must achieve the stated levels of reliability and uptime, and (iii) the total owning and operating costs must be kept to a competitive minimum.


2.1 The single-channel description

The semiclassical theory is used to estimate the fusion cross section in the onedimensional potential model which assumes that one can describe the degree of freedom only of the relative motion between the colliding heavy ions [19, 20]. The semiclassical theory deals with the Schrödinger equation assuming independent energy and angular momentum and the potential energy for the radial part of the relative motion through quantum tunneling:

$$\left[-\hbar^2\nabla^2/2\mu + V(r) - E\right]\Psi(\mathbf{r}) = \mathbf{0} \tag{1}$$

where μ is the reduced mass of the system and V rð Þ is the total potential energy of the system. Semiclassical reaction amplitudes can be evaluated as a function of time, assuming the particle trajectory to be determined by classical dynamics, including Coulomb and real nuclear and centrifugal potentials, which can be written in the form

$$V(r) = V\_C(r) + V\_N(r) + V\_l(r) \tag{2}$$

In coupled-channel effects on the elastic channel, the imaginary part should be added to the nuclear potential, represented by complex potential as

$$V\_N(r) = U\_N(r) - iW(r) \tag{3}$$

Fusion Reaction of Weakly Bound Nuclei DOI: http://dx.doi.org/10.5772/intechopen.80582

4 He+7

systems 6

Li with separation energy Sα = 8.664 MeV and 12C breakup into three α

Be.

The semiclassical theory is used to estimate the fusion cross section in the onedimensional potential model which assumes that one can describe the degree of freedom only of the relative motion between the colliding heavy ions [19, 20]. The semiclassical theory deals with the Schrödinger equation assuming independent energy and angular momentum and the potential energy for the radial part of the

where μ is the reduced mass of the system and V rð Þ is the total potential energy of the system. Semiclassical reaction amplitudes can be evaluated as a function of time, assuming the particle trajectory to be determined by classical dynamics, including Coulomb and real nuclear and centrifugal potentials, which can be

In coupled-channel effects on the elastic channel, the imaginary part should be

added to the nuclear potential, represented by complex potential as

=2μ þ V rð Þ� E Ψð Þ¼ r 0 (1)

V rð Þ¼ VCð Þþ r VNð Þþ r Vlð Þr (2)

VNð Þ¼ r UNð Þ� r iW rð Þ (3)

channel is described by the continuum discretized coupled-channel (CDCC) method. The continuum that describes the breakup channel is discretized into bins [13, 14]. To study the coupled-channel problem, this requires a profound truncation of the continuum into discrete bin of energy into equally spaced states. The CDCC method is totally based on this concept. Surrey group extended the discretization procedure discussed in [14] for the deuteron case to study the breakup and fusion reactions of systems involving weakly bound nuclei [15, 16]. Recently, Majeed and Abdul-Hussien [17] utilized the semiclassical approach based on the theory of Alder and Winther. They carried out their calculations to investigate the role of the breakup channel on the fusion cross section σfus and fusion barrier distribution Dfus for 6,8H halo [17]. Semiclassical coupled-channel calculations in heavy-ion fusion reactions for the systems 40Ar + 110Pd and 132Sn + 48Ca were carried out by Majeed et al. [18]. They argued that including the channel coupling between the elastic channel and the continuum enhances the fusion reaction cross section σfus and the fusion barrier distribution Dfus calculations quite well below and above the Coulomb barrier for medium and heavy systems. This study aims to employ a semiclassical approach by adopting Alder and Winther (AW) [19] theory originally used to treat the Coulomb excitation of nuclei. The semiclassical approach has been implemented and coded using FORTRAN programming codenamed (SCF) which is written and developed by Canto et al. [20]. The fusion cross section σfus and fusion barrier distribution Dfus are calculated here utilizing the semiclassical approach. The results from the present study are compared with the quantum mechanical calculations using the FORTRAN code (CC) [21] and with the experimental data for the three

Nuclear Fusion - One Noble Goal and a Variety of Scientific and Technological Challenges

) 3α [3, 11, 12]. The breakup

particles induced by neutrons or protons by 12C (p, p0

Li + 64Ni, 11B + 159Tb, and 12C + 9

2. The semiclassical theory

written in the form

22

2.1 The single-channel description

relative motion through quantum tunneling:

�ℏ2 ∇2

The method can be extended to describe interference of different l waves due to strong nuclear attraction and absorption caused by the imaginary nuclear potential [6, 20, 22]. When the two nuclei come across the potential barrier into the inner region, the fusion occurs according to the semiclassical theory, and the penetrability probability below barrier can be evaluated usingWKB approximation [5, 6, 19, 23, 24]:

$$P\_{fus}^{WKB}(l,E) = \frac{1}{\mathbf{1} + \exp\left[2\int\_{r\_b^{(l)}}^{r\_s^{(l)}} \kappa\_l(r) dr\right]}\tag{4}$$

Then, the latter can be rewritten as follows:

$$P\_{fus}^{\rm WKB}(l, E) = \frac{1}{1 + \exp\left[\frac{2\pi}{\hbar\Omega\_l}(V\_b(l) - E)\right]}\tag{5}$$

where κlð Þr is the local wave number and r ð Þl b and r ð Þl a are the inner and outer classical turning points at the fusion barrier potential. If one approximates the fusion barrier by a parabolic function, then the penetrability probability above barrier is given by the Hill-Wheeler formula [3, 19]:

$$P\_{\rm fus}^{\rm WH}(l, E) = \frac{1}{\mathbf{1} + \exp\left[\frac{2\pi}{\hbar \Omega\_l} (E - \mathbf{V}\_b(l))\right]} \tag{6}$$

where Vbð Þl and Ωl are the height and the curvature parameter of the fusion barrier for the partial wave, respectively, and E is the bombarding energy. Ignoring the l dependence of ω and of the barrier position Rb and assuming that the l dependence of Vbð Þl is given only by the difference of the centrifugal potential energy, one can obtain Wong's formula which is given in Section 5. The fusion cross sections can be estimated by the one-dimensional WKB approximation by the following relations [21, 24]:

$$
\sigma\_{\rm fus}(E) = \frac{\pi}{\kappa^2} \sum (2l+1) P\_{\rm fus}^{\rm MVKB}(l, E) \tag{7}
$$

$$P\_{fus}^{\prime}(l,E) = \frac{4k}{E} \int dr \left| u\_{\eta l}(k\_{\gamma}, r) \right|^2 \mathcal{W}\_{fus}^{\prime}(r) \tag{8}$$

where uγl kγ ;r � � represents the radial wave function for the partial wave l in channel γ and Wγ fusð Þr is the absolute value of the imaginary part of the potential associated to fusion in that channel.

The complete fusion cross section in heavy ions evaluated using semiclassical theory is based on the classical trajectory approximation r. And the relevant intrinsic degrees of freedom of the projectile, represented by ξ with applying the continuum discretized coupled-channel (CDCC) approximation of Alder and Winther (AW) theory [16], have been proposed [17]. The projectile Hamiltonian is then given by

$$h = h\_0(\xi) + V(\xi, r) \tag{9}$$

where h0ð Þξ is the intrinsic Hamiltonian and Vð Þ¼ ξ;r VNð Þþ ξ;r VCð Þ ξ;r , Vð Þ ξ;r is the interaction between the projectile and target nuclei. The Rutherford trajectory depends on the collision energy, E, and the angular momentum, l. In this case the trajectory is the solution of the classical motion equations with the potential

V rð Þ¼ h i Ψ0jV rð Þj ; ξ Ψ0 , where Ψ0 is the ground state (g.s.) of the projectile. In this way, the interaction becomes time-dependent in the ξ-space Vlð Þ¼ ξ:t V rl tð Þ; ξ , and the eigenstates of the intrinsic Hamiltonian jΨγ satisfy the Schrödinger equation [25, 26]:

$$h|\Psi\_{\mathcal{I}}\rangle = \varepsilon|\Psi\_{\mathcal{I}}\rangle\tag{10}$$

ψ ξð Þ¼ ; t ∑

α

σβ ¼ π k2 ∑ ℓ

σF ¼ ∑ β

PF ℓð Þ¼ β

tial associated to fusion in that channel.

approach on the classical trajectory and Tð Þ β

π k2 ∑ ℓ

4k E ð WF

PF ℓð Þ β ≃P

obtains the AW equations [27]

Fusion Reaction of Weakly Bound Nuclei DOI: http://dx.doi.org/10.5772/intechopen.80582

ℓ ¼ aβð Þ ℓ; t ! þ∞ � � �

expansions, we get [28]

with

channel β, and WF

approximation [27]

ð Þ β

tial barrier in channel β [19].

where P

That is [19],

25

Pð Þ β

iℏa\_ βð Þ¼ ℓ; t ∑

�

β

and inserting this expansion into the Schrödinger equation for ψ ξð Þ ; t , one

aγ ð Þ ℓ; t φβjVℓð Þj ξ; t φβ � �e

2 and the angle-integrated cross section is [19]

ð Þ 2ℓ þ 1 Pð Þ β

ð Þ 2ℓ þ 1 PF

β ð Þr uβℓ kβ;r � � � � �

ℓ is the probability that the system is in channel β at the point of closest

ℓ Eβ

β is the absolute value of the imaginary part of the optical poten-

Above, uβℓ kβ;r � � represents the radial wave function for the ℓth partial wave in

To use the AW method to evaluate the fusion cross section, we make the

ð Þ β ℓ Tð Þ β ℓ Eβ

with energy Eβ ¼ E � εβ and reduced mass μ ¼ MPMT=ð Þ MP þ MT , where MP, MT are the masses of the projectile and target, respectively, tunnels through the poten-

We now proceed to study the CF cross sections in reactions induced by weakly bound projectiles. For simplicity, we assume that the g.s. is the only bound state of the projectile and that the breakup process produces only two fragments, F1 and F2. In this way, the labels β ¼ 0 and β 6¼ 0 correspond, respectively, to the g.s. and the breakup states represented by two unbound fragments. Neglecting any sequential contribution, the CF can only arise from the elastic channel. In this way, the cross section σCF can be obtained from Eq. (20), dropping contributions from β 6¼ 0.

ℓð Þ β

� 2

� � (18)

To extend this method to fusion reactions, we start with the quantum mechanical calculation of the fusion cross section in a coupled-channel problem. For simplicity, we assume that all channels are bound and have zero spin. The fusion cross section is a sum of contributions from each channel. Carrying out partial-wave

These equations are solved with the initial conditions aβð Þ¼ ℓ; t ! �∞ δβ0, which means that before the collision ð Þ t ! �∞ , the projectile was in its ground state. The final population of channel β in a collision with angular momentum ℓ is

aβð Þ ℓ; t φβð Þξ e

�iεβ t=ℏ (15)

�ið Þ εβ�εγ t=ℏ (16)

ℓ (17)

dr (19)

� � (20)

� � is the probability that a particle

After expanding the wave function in the basis of intrinsic eigenstates

$$\Psi(\xi, t) = \sum a\_{\gamma}(l, t)\Psi\_{\gamma}(\xi)e^{-ic\_{\gamma}t/\hbar} \tag{11}$$

and inserting Eq. (11) into the Schrödinger equation for Ψð Þ ξ; t , the AW equations can be obtained:

$$i\hbar \,\dot{a}\_{\gamma}(l,t) = \sum\_{\varepsilon} \left< \Psi\_{\gamma} | V(\xi, t) | \Psi\_{\gamma} \right> e^{i(\varepsilon\_{\gamma} - \varepsilon\_{\varepsilon})t/\hbar} \gamma e(l, t) \tag{12}$$

These equations should be solved with initial conditions aγ ð Þ¼ l; t ! �∞ δγ0 which mean that before the collision ð Þ t ! �∞ , the projectile was in its ground state. The final population of the channel γ in a collision with angular momentum l is Pγ fusð Þ¼ l; E aγ ð Þ l; t ! �∞ 2 . Eq. (8) gives the general expression for the fusion cross section in multichannel scattering [26].

#### 2.2 The coupled-channel description

The variables employed to describe the collision are the projectile-target separation vector r ! and the relevant intrinsic degrees of freedom of the projectile ξ. For simplicity, we neglect the internal structure of the target. The Hamiltonian is then given by [27]

$$H = H\_0(\xi) + V(\overrightarrow{r}, \xi) \tag{13}$$

where H0ð Þξ is the intrinsic Hamiltonian of the projectile and Vðr ! , ξÞ represents the projectile-target interaction. Since the main purpose of the present work is to test the semiclassical model in calculations of sub-barrier fusion, the nuclear coupling is neglected. Furthermore, for the present theory-theory comparison, only the Coulomb dipole term is taken into account. The eigenvectors of H0ð Þξ are given by the equation [27]

$$H\_0|\rho\_\beta\rangle = \varepsilon\_\beta|\rho\_\beta\rangle\tag{14}$$

where εβ is energy of internal motion. The AW method is implemented in two steps. First, one employs classical mechanics for the time evolution of the variable r !. The ensuing trajectory depends on the collision energy, E, and the angular momentum, l. In its original version, an energy symmetrized Rutherford trajectory rℓ ! ð Þt was used. In our case, the trajectory is the solution of the classical equations of motion with the potential Vðr !Þ ¼ ⟨φ0|Vðr !, ξÞφ0⟩, where jφ0i is the ground state of the projectile. In this way, the coupling interaction becomes a time-dependent interaction in the ξ-space, Vℓð Þ� ξ; t Vðrℓ ! ð Þt , ξÞ. The second step consists in treating the dynamics in the intrinsic space as a time-dependent quantum mechanics problem. Expanding the wave function in the basis of intrinsic eigenstates [19]

Fusion Reaction of Weakly Bound Nuclei DOI: http://dx.doi.org/10.5772/intechopen.80582

$$\Psi(\xi, t) = \sum\_{\beta} a\_{\beta}(\ell, t) \rho\_{\beta}(\xi) e^{-i\epsilon\_{\beta}t/\hbar} \tag{15}$$

and inserting this expansion into the Schrödinger equation for ψ ξð Þ ; t , one obtains the AW equations [27]

$$i\hbar \dot{\mathbf{a}}\_{\beta}(\ell, \mathbf{t}) = \sum\_{a} \mathbf{a}\_{\gamma}(\ell, \mathbf{t}) \langle \boldsymbol{\uprho}\_{\beta} | \mathbf{V}\_{\ell}(\xi, \mathbf{t}) | \boldsymbol{\uprho}\_{\beta} \rangle e^{-i \left(\boldsymbol{\uprho} - \boldsymbol{\uprho}\_{\gamma}\right) \mathbf{t}/\hbar} \tag{16}$$

These equations are solved with the initial conditions aβð Þ¼ ℓ; t ! �∞ δβ0, which means that before the collision ð Þ t ! �∞ , the projectile was in its ground state. The final population of channel β in a collision with angular momentum ℓ is Pð Þ β ℓ ¼ aβð Þ ℓ; t ! þ∞ � � � � 2 and the angle-integrated cross section is [19]

$$
\sigma\_{\beta} = \frac{\pi}{k^2} \sum\_{\ell} (2\ell + 1) P\_{\ell}^{(\beta)} \tag{17}
$$

To extend this method to fusion reactions, we start with the quantum mechanical calculation of the fusion cross section in a coupled-channel problem. For simplicity, we assume that all channels are bound and have zero spin. The fusion cross section is a sum of contributions from each channel. Carrying out partial-wave expansions, we get [28]

$$\sigma\_{\boldsymbol{F}} = \sum\_{\boldsymbol{\beta}} \left[ \frac{\boldsymbol{\pi}}{k^2} \sum\_{\ell} (2\ell + \mathbf{1}) P\_{\ell}^{\boldsymbol{F}}(\boldsymbol{\beta}) \right] \tag{18}$$

with

V rð Þ¼ h i Ψ0jV rð Þj ; ξ Ψ0 , where Ψ0 is the ground state (g.s.) of the projectile. In this way, the interaction becomes time-dependent in the ξ-space Vlð Þ¼ ξ:t V rl tð Þ; ξ ,

Nuclear Fusion - One Noble Goal and a Variety of Scientific and Technological Challenges

hjΨγ ¼ εjΨγ

After expanding the wave function in the basis of intrinsic eigenstates

Ψð Þ¼ ξ; t ∑aγ ð Þ l; t Ψγ ð Þξ e

and inserting Eq. (11) into the Schrödinger equation for Ψð Þ ξ; t , the AW

Ψγ jVð Þj ξ; t Ψγ e

These equations should be solved with initial conditions aγ ð Þ¼ l; t ! �∞ δγ0 which mean that before the collision ð Þ t ! �∞ , the projectile was in its ground state. The final population of the channel γ in a collision with angular momentum l

The variables employed to describe the collision are the projectile-target separa-

simplicity, we neglect the internal structure of the target. The Hamiltonian is then

H ¼ H0ð Þþ ξ V r!; ξ

the projectile-target interaction. Since the main purpose of the present work is to test the semiclassical model in calculations of sub-barrier fusion, the nuclear coupling is neglected. Furthermore, for the present theory-theory comparison, only the Coulomb dipole term is taken into account. The eigenvectors of H0ð Þξ are given by

¼ εβjφβ

where εβ is energy of internal motion. The AW method is implemented in two steps. First, one employs classical mechanics for the time evolution of the variable

! ð Þt was used. In our case, the trajectory is the solution of the classical equations of

treating the dynamics in the intrinsic space as a time-dependent quantum mechanics problem. Expanding the wave function in the basis of intrinsic eigenstates [19]

where H0ð Þξ is the intrinsic Hamiltonian of the projectile and Vðr

H0jφβ

!. The ensuing trajectory depends on the collision energy, E, and the angular momentum, l. In its original version, an energy symmetrized Rutherford trajectory

the projectile. In this way, the coupling interaction becomes a time-dependent

!Þ ¼ ⟨φ0|Vðr

! and the relevant intrinsic degrees of freedom of the projectile ξ. For

ið Þ εγ�εϵ t

. Eq. (8) gives the general expression for the fusion

satisfy the Schrödinger

(10)

�iεγ t=ℏ (11)

=ℏ γϵð Þ l; t (12)

(13)

! , ξÞ represents

(14)

!, ξÞφ0⟩, where jφ0i is the ground state of

! ð Þt , ξÞ. The second step consists in

and the eigenstates of the intrinsic Hamiltonian jΨγ

iℏ a\_ γ ð Þ¼ l; t ∑

 2 ϵ

equation [25, 26]:

equations can be obtained:

fusð Þ¼ l; E aγ ð Þ l; t ! �∞

2.2 The coupled-channel description

cross section in multichannel scattering [26].

is Pγ

tion vector r

given by [27]

the equation [27]

motion with the potential Vðr

interaction in the ξ-space, Vℓð Þ� ξ; t Vðrℓ

r

rℓ

24

$$P\_{\ell}^{F}(\beta) = \frac{4k}{E} \int \mathcal{W}\_{\beta}^{F}(r) |u\_{\beta \ell}(k\_{\beta}, r)|^{2} dr \tag{19}$$

Above, uβℓ kβ;r � � represents the radial wave function for the ℓth partial wave in channel β, and WF β is the absolute value of the imaginary part of the optical potential associated to fusion in that channel.

To use the AW method to evaluate the fusion cross section, we make the approximation [27]

$$P\_{\ell}^{\mathrm{F}}(\boldsymbol{\beta}) \simeq \overline{\mathbf{P}}\_{\ell}^{(\boldsymbol{\beta})} T\_{\ell}^{(\boldsymbol{\beta})} \left( \mathbf{E}\_{\boldsymbol{\beta}} \right) \tag{20}$$

where P ð Þ β ℓ is the probability that the system is in channel β at the point of closest approach on the classical trajectory and Tð Þ β ℓ Eβ � � is the probability that a particle with energy Eβ ¼ E � εβ and reduced mass μ ¼ MPMT=ð Þ MP þ MT , where MP, MT are the masses of the projectile and target, respectively, tunnels through the potential barrier in channel β [19].

We now proceed to study the CF cross sections in reactions induced by weakly bound projectiles. For simplicity, we assume that the g.s. is the only bound state of the projectile and that the breakup process produces only two fragments, F1 and F2. In this way, the labels β ¼ 0 and β 6¼ 0 correspond, respectively, to the g.s. and the breakup states represented by two unbound fragments. Neglecting any sequential contribution, the CF can only arise from the elastic channel. In this way, the cross section σCF can be obtained from Eq. (20), dropping contributions from β 6¼ 0. That is [19],

Nuclear Fusion - One Noble Goal and a Variety of Scientific and Technological Challenges

$$
\sigma\_{\rm CF} = \frac{\pi}{k^2} \sum\_{\ell} (2\ell + \mathbf{1}) P\_{\ell}^{\rm Surv} T\_{\ell}^{(0)}(E) \tag{21}
$$

4. Results and discussion

Fusion Reaction of Weakly Bound Nuclei DOI: http://dx.doi.org/10.5772/intechopen.80582

barrier are displayed in Table 1.

4.1 The reaction 6

12C + 9

6

6

Table 1.

Figure 1.

fusion reaction barrier distribution Dfus (mb/MeV).

6

27

12C + 9

In this section, the theoretical calculations are obtained for total fusion reaction σfus, and the fusion barrier distribution Dfus using the semiclassical theory adopted the continuum discretized coupled channel (CDCC) to describe the effect of the breakup channel on fusion processes. The semiclassical calculations are carried out

Be. The values of the height Vc, radius Rc, and curvature ℏω for the fusion

The calculations of the fusion cross section σfus and fusion barrier distribution Dfus are presented in Figure 1 panel (a) and panel (b), respectively, for the system

The experimental data for this system are obtained from Ref. [32]. The real and

Li + 64Ni. The dashed blue and red curves represent the semiclassical and full quantum mechanical calculations without coupling, respectively. The solid blue and red curves are the calculations including the coupling effects for the semiclassical and full quantum mechanical calculations, respectively. Panel (a) shows the comparison between our semiclassical and full quantum mechanical calculations with

imaginary Akyüz-Winther potential parameters obtained by using chi-square

System Vc Rc ℏω Refs

Li + 64Ni 12:41 9:1 3:9 [32] 11B + 159Tb 40:34 10:89 4:42 [33]

The fusion barrier parameters are height Vc ð Þ MeV , radius Rcð Þ fm , and curvature ℏω ð Þ MeV :

Be 4.28 7.43 2.61 [34]

The comparison of the coupled-channel calculations of semiclassical treatment (red curves) and full quantum mechanical (blue curves) with the experimental data of complete fusion (black-filled circles) [32] for

Li + 64Ni system. Panel (a) refers to the total fusion reaction cross section σfus (mb), and panel (b) provides the

Li + 64Ni, 11B + 159Tb and

using the (SCF) code, while the full quantum mechanical calculations are

performed by using the code (CC) for the systems 6

Li + 64Ni

the respective experimental data (solid circles).

where

$$P\_{\ell}^{\text{Surv}} \equiv \overline{P}\_{\ell}^{(0)} = |a\_0(\ell, t\_{ct})|^2 \tag{22}$$

is usually called survival (to breakup) probability [19].