DOE-HDBK-1122-2009 Chg Notice 2, Radiological Control Technician Training (Part 3 of 9)

Part 3 of 9 Radiological Control Technician Training Fundamental Academic Training Instructor’s Guide Phase I Coordinated and Conducted for the Office of Health, Safety and Security U.S. Department of Energy DOE-HDBK-1122-2009 Radiological Control Technician Instructor’s Guide This page intentionally left blank. 1.01- ii DOE-HDBK-1122-2009 Radiological Control Technician Instructor’s Guide Table of Contents Page Module 1.01 Basic Mathematics and Algebra……………………………………………... 1.01-1 Module 1.02 Unit Analysis and Conversion……………………………………………….. 1.02-1 Module 1.03 Physical Sciences……………………………………………………………..1.03-1 Module 1.04 Nuclear Physics……………………………………………………………… 1.04-1 Module 1.05 Sources of Radiation………………………………………………………….1.05-1 Module 1.06 Radioactivity and Radioactive Decay……………………………………….. 1.06-1 Module 1.07 Interaction of Radiation with Matter………………………………………… 1.07-1 Module 1.08 Biological Effects of Radiation……………………………………………… 1.08-1 Module 1.09 Radiological Protection Standards…………………………………………... 1.09-1 Module 1.10 ALARA……………………………………………………………………… 1.10-1 Module 1.11 External Exposure Control…………………………………………………... 1.11-1 Module 1.12 Internal Exposure Control…………………………………………………… 1.12-1 Module 1.13 Radiation Detector Theory…………………………………………………... 1.13-1 1.01-iii DOE-HDBK-1122-2009 Radiological Control Technician Instructor’s Guide This page intentionally left blank. 1.01-iv DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide Course Title: Radiological Control Technician Module Title: Basic Mathematics and Algebra Module Number: 1.01 Objectives: 1.01.01 Add, subtract, multiply, and divide fractions. 1.01.02 Add, subtract, multiply, and divide decimals. 1.01.03 Convert fractions to decimals and decimals to fractions. 1.01.04 Convert percent to decimal and decimal to percent. 1.01.05 Add, subtract, multiply, and divide signed numbers. 1.01.06 Add, subtract, multiply, and divide numbers with exponents. 1.01.07 Find the square roots of numbers. 1.01.08 Convert between numbers expressed in standard form and in scientific notation. 1.01.09 Add, subtract, multiply, and divide numbers expressed in scientific notation. 1.01.10 Solve equations using the "Order of Mathematical Operations." 1.01.11 Perform algebraic functions. 1.01.12 Solve equations using common and/or natural logarithms. References: 1. DOE-HDBK-1014/1-92 (June 1992) "Mathematics: Volume 1 of 2"; DOE Fundamentals Handbook Series. Instructional Aids: 1. Overheads 2. Overhead projector/screen 3. Chalkboard/whiteboard 4. Lessons learned 1.01 - 1 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide I. MODULE INTRODUCTION A. Self Introduction 1. Name 2. Phone number 3. Background 4. Emergency procedure review B. Motivation Radiological control operations frequently require the RCT to use arithmetic and algebra to perform various calculations. These include scientific notation, unit analysis and conversion, radioactive decay calculations, dose rate/distance calculations, shielding calculations, and stay-time calculations. A good foundation in mathematics and algebra is important to ensure that the data obtained from calculations is accurate. Accurate data is crucial to the assignment of proper radiological controls. C. Overview of Lesson 1. Fractions 2. Decimals 3. Percent 4. Signed Numbers 5. Exponents 6. Square Roots 7. Scientific Notation 8. Order of Mathematical Operations

9. Algebra 10. Logarithms O.H.: Objectives D. Introduce Objectives 1.01 - 2 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide II. MODULE OUTLINE A. Symbols for Basic Operations See Table 1. - 1. The four basic mathematical operations are addition, subtraction, "Symbols for Basic multiplication, and division. Furthermore, it is often necessary to Mathematical group numbers or operations using parentheses or brackets. In Operations" writing problems in this course the notation shown in Table 1 is used to denote the operation to be performed on the numbers. Objective 1.01.01 B. Fractions 1. Whole numbers consist of the normal counting numbers and zero. {e.g., 0, 1, 2, 3, 4...} 2. A fraction is part of a whole number. It is simply an expression of a division of two whole numbers. A fraction is written in the format: a or a/b b a. The number above the bar a is called the numerator and the number below the bar b is called the denominator. A proper fraction is a fraction in which the number in the numerator is less than the number in the denominator. If the numerator is greater than the denominator then it is an improper fraction. For example, ½ and ¼ are proper fractions, while, 25/5, 15/7, or 61/27 are improper fractions. b. Any whole number can be written as a fraction by letting the whole number be the numerator and 1 be the denominator. For example: 5 2 05 = 2 = 0 = 1 1 1 c. Five can be written as 10/2, 15/3, 20/4, etc. Similarly, the fraction ¼ can be written as 2/8, 3/12, 4/16, etc. These are called equivalent fractions. An equivalent fraction is built up, per se, by multiplying the numerator and the denominator by the same non-zero number. For example: 1.01 - 3 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 3 3×2 6 3 3×5 15 = = = = 4 4 ×2 8 4 4 ×5 20 d. A fraction is reduced by dividing the numerator and the denominator by the same non-zero number. For example: 12 12 ÷2 6 6 6 ÷3 2 = = = = 18 18 ÷ 2 9 9 9 ÷3 3 e. A fraction is reduced to lowest terms when 1 is the only number that divides both numerator and denominator evenly. This is done by finding the greatest common multiple between the numerator and denominator. In the previous example, two successive reductions were performed. For the fraction 12/18, the greatest common multiple would be 6, or (2 × 3), which results in a reduction down to a denominator of 3. f. A whole number written with a fraction is called a mixed number. Examples of mixed numbers would be 1½, 3¼, 5¾, etc. A mixed number can be simplified to a single improper fraction using the following steps: 1) Multiply the whole number by the denominator of the fraction. 2) Add the numerator of the fraction to the product in step 1. 2) Add the numerator of the fraction to the product in step 1. 3) Place the sum in step 2 as the numerator over the denominator. 4) Example: ( )× + 3 23 3 5 4 5 = = 4 4 4 3. Adding and Subtracting Fractions a. Fractions with the Same Denominator 1) To add two fractions which have the same denominator: 1.01 - 4 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide a) Add the numerators b) Place the sum of step 1 over the common denominator c) Reduce fraction in step 2 to lowest terms (if necessary) 2) Example: 1 3 1+ 3 4 + = = 5 5 5 5 3) Subtraction of two fractions with the same denominator is accomplished in the same manner as addition. For example:

5 3 5 −3 2 2 ÷ 2 1 − = = = = 8 8 8 8 8 ÷ 2 4 b. Fractions with Different Denominators 1) To add two fractions with different denominators requires that the fractions be built up so that they have the same denominator. This is done by finding the lowest common denominator. Once a common denominator is obtained, the rules given above for the same denominator apply. 2) For example, 1/3 + 2/5. The fraction 1/3 could be built up to 2/6, 3/9, 4/12, 5/15, 6/18, 7/21, etc. The fraction 2/5 could be built up to 4/10, 6/15, 8/20, 10/25, etc. The lowest common denominator for the two fractions would be 15. The problem would be solved as follows: 1 2 1×5 2×3 5 6 11 + = + = + = 3 5 3×5 5×3 15 15 15 3) Subtraction of fractions with different denominators is accomplished using the same steps as for addition. For example: 3 2 3×3 2×4 9 8 1 − = − = − = 4 3 4×3 3×4 12 12 12 1.01 - 5 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 4. Multiplying and Dividing Fractions a. Multiplication of fractions is much easier than addition and subtraction, especially if the numbers in the numerators and denominators are small. Fractions with larger numerators and/or denominators may require additional steps. In either case, the product of the multiplication will most likely need to be reduced in order to arrive at the final answer. To multiply fractions: 1) Multiply the numerators. 2) Multiply the denominators. 3) Place product in step 1 over product in step 2. 4) Reduce fraction to lowest terms. 5) Example: 5 3 5×3 15 15 ÷3 5 × = = = = 6 4 6×4 24 24 ÷3 8 b. A variation on the order of the steps to multiply fractions is to factor the numerators and denominators first, reduce and cancel, and then multiply. For example: 3 20 3 2× ×2 5 3 2 2 5 5× × × 5 × = × = = = 8 9 × × 3×3 2 2 2 3 3 2×32 2 2 × × × × 6 5. Reciprocals a. Two numbers whose product is 1 are called reciprocals, or multiplicative inverses. For example: 1 1 1) 5 and are reciprocals because 5× =1 5 5 4 5 4 5 2) and are reciprocals because × =1 5 4 5 4 3) 1 is its own reciprocal because 1 1× =1 4) 0 has no reciprocal because 0 times any number is 0 not 1 b. The symbol for the reciprocal, or multiplicative inverse, of a 1 non-zero real number a is . a 1.01 - 6 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide Every real number except 0 has a reciprocal. Therefore, for every non-zero real number a, there is a unique real number 1 such that a 1 a × =1 a c. Now, look at the following product: ⎛ 1 1 ⎞ ⎛ 1 ⎞⎛ 1 ⎞( )⎜ ⎟ ⎜ a × ⎟⎜b× ⎟ =1 1 =1ab × = × ⎝ a b ⎠ ⎝ a ⎠⎝ b ⎠ 6. Relationship of multiplication to division a. The operation of division is really just inverted multiplication (reciprocals). Notice from the above examples the reciprocal of a fraction is merely "switching" the numerator and denominator. The number 5 is really 5/1, and the reciprocal of 5 is 1/5. Likewise, the reciprocal of 2/3 is 3/2. b. Fractions are a division by definition. Division of fractions is accomplished in two steps: 1) Invert the second fraction, i.e., change it to its reciprocal, and change the division to multiplication. 2) Multiply the two fractions using the steps stated above. 3) Examples: (1/2)/3 = 1 / (2/3) = 1 x 3/2 = 3/2 C. Decimals

1. A decimal is another way of expressing a fraction or mixed number. It is simply the numerical result of division (and fractions are division). Recall that our number system is based on 10 ("deci" in "decimal" means ten) and is a place-value system; that is, each digit {i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8, 9} in a numeral has a particular value determined by its location or place in the number. For a number in decimal notation, the numerals to the left of the decimal point comprise the whole number, and the numerals to the right are the decimal fraction, (with the denominator being a power of ten). Objective 1.01.02 See Fig. 1 - "Decimal Places" 1.01 - 7 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 2. For example, the numeral 125.378 (decimal notation) represents the expanded numeral: 3 7 8100 + 20 + 5 + + + 10 100 1000 3. If this numeral were written as a mixed number, we would write it as: 378125 1000 4. Addition and Subtraction of Decimals a. In order to add or subtract decimals use the following steps: 1) Arrange the numbers in a column so that the decimal points are aligned. 2) Add or subtract in columns from right to left. (Additional Subtrahend - a zeros may need to be added to the right of the number that is to be subtrahend.) subtracted from the minuend). 3) Place the decimal point in the answer in line with the other decimal points. 4) Examples: 21.3 654.200 +4.2 -26.888 25.5 627.312 5. Multiplying Decimals a. To multiply decimal numbers, do the following: 1) Multiply the numbers as if there were no decimal points. 2) Count the number of decimal places in each number and add them together. 3) Place the decimal point in the product so that it has the same number of decimal places as the sum in step 2. 1.01 - 8 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 4) Examples: 5.28 ×3.7 3696 1584 19.536 0.04 ×0.957 028 020 036 000 0.03828 6. Division of Decimals a. The steps for division of decimals are as follows: 1) Move the decimal point of the divisor to the right until it becomes a whole number. 2) Move the decimal point of the dividend to the right the same number of places moved in step 1. Divisor - The number by which a dividend is divided. Dividend - a number to be divided. 3) Divide the numbers as if they were whole numbers. 4) Example: 28 0 200 200 50 0.25 7.00 7. Decimal Forms a. As we have learned, decimals are the result of division (or a fraction). When the remainder of the division is zero, the resulting decimal is called a terminating or finite decimal. For example, fractions like 2/5, 3/8, and 5/6 will all result in finite decimals. These fractions and the resulting decimals are known as rational numbers. 1.01 - 9 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide b. On the other hand, fractions like 1/3, 2/7, 5/11, and 7/13 result in a non-terminating or infinite decimal. For example, 2/7 results in the decimal 0.285714286 . . . , the dots meaning that the decimal continues without end. These numbers are known as irrational numbers. Note that even though irrational numbers are non-terminating, some (e.g., 1/3 and 5/11) are repeating or periodic decimals because the same digit or block of digits repeats unendingly. For example: 1 5 = 0.3333... = 0.454545... 3 11 c. A bar is often used to indicate the block of digits that repeat, as shown below:

1 5 = 0.3 = 0.45 3 11 D. Fraction to Decimal Conversion Objective 1.01.03 To convert a fraction to a decimal we simply perform the operation of division that the fraction represents. For example, the fraction 3/4 represents "3 divided by 4," and would be converted as follows: 0.75 4 3.00 28 20 20 0 E. Percent Objective 1.01.04 1. Percentage is a familiar and widely used concept for expressing common and decimal fractions. Most people know the meaning of terms such as 100 percent and 50 percent. The word percent actually means "out of a hundred." (Consider that there are 100 "cents" in a dollar, and that a "century" is 100 years.) A percent is simply a fraction whose denominator is 100. Thus, 50 percent means 50/100, or 0.50, while 25 percent means 25/100, or 0.25. 2. Percent is abbreviated by the symbol %. So, 75 percent is written 75%. 3. Converting Decimal to Percent 1.01 - 10 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide a. A decimal fraction is changed to a percent by moving the decimal point two places to the right and adding a percent sign. For example, 1/8 equals 0.125. Therefore: 1 0.125 = 12.5%= rr8 b. A percent is changed to a common fraction by omitting the percent sign, placing the number over 100, and reducing the resulting fraction if possible. For example, 32% equals 32/100 which reduces to 8/25. When the percent consists of a mixed decimal number with a percent sign, the resulting fraction will contain a mixed decimal numerator. This can be changed to a whole number by multiplying the numerator and the denominator by 10, 100, 1,000, etc. For example: 40.25 40.25×100 4025 80540.25% = = = = 100 100×100 10,000 2000 c. Percentage is most frequently used to indicate a fractional part. Thus 20% of the total power output for 75% of the employees refer to fractional parts of some total number. To perform arithmetic operations with a percent, it is normally changed to a common or decimal fraction. Usually, a decimal fraction is more convenient. 4. Converting Percent to Decimal a. A percent is changed to a decimal fraction by omitting the percent sign and moving the decimal point two places to the left. For example: 48% = 0.48ss b. Thus, 92% equals 0.92, 8% equals 0.08, and so on. F. Signed Numbers 1. The numbers that are used to quantify the number of objects in a group, the "counting numbers," are always positive numbers; that is, they are always greater than zero. However, there are many occasions when negative numbers (numbers less than zero) must be used. These numbers arise when we try to describe measurement in a direction opposite to the positive numbers. For example, if we assign a value of +3 to a point which is 3 feet above the ground, what number should be assigned to a point which is 3 feet below the ground? Perhaps the most familiar example of the use of negative numbers is the measurement of temperature, where temperatures below an arbitrary reference level are assigned negative values. 1.01 - 11 Objective 1.01.05 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 2. Every number has a sign associated with it. The plus (+) sign indicates a positive number, whereas the minus (-) sign indicates a negative number. When no sign is given, a plus sign (+) is implied. The fact that the plus and minus signs are also used for the arithmetic operations of addition and subtraction should not be a cause for confusion, for we shall see that they have equivalent meanings.

3. Every number has an absolute value, regardless of its sign. The absolute value indicates the distance from zero, without regard to direction. The number 5 is 5 units from zero, in the positive direction. The number -5 is also 5 units from zero, but in the negative direction. The absolute value of each of these numbers is 5. The absolute value of a number is indicated by a pair of vertical lines enclosing the number: 5 . 4. Operations with Signed Numbers See Fig. 2 - a. The arithmetic operations of addition, subtraction, "Number Line" multiplication, and division of signed numbers can be more easily visualized if the numbers are placed on a number line. The positive numbers are greater than zero, and they lie to the right of zero on the number line. The negative numbers are less than zero, and lie to the left of zero on the number line. b. The number line extends an infinite distance in each direction and therefore includes all numbers. The process of addition can be considered as counting in one direction or the other from a starting point on the number line. For example, let us add 1 + 2. We locate +1 on the number line and then count 2 units to the right, since we are adding +2. The result will be +3. To illustrate further, let us add +2 and -4. We first locate +2 on the number line and then count 4 units to the left. We end up at -2. c. The number line is useful for illustrating the principles of addition, but it clearly would be inconvenient to use in the case of large numbers. Consequently, the following rules were developed to govern the addition process. 5. Adding and Subtracting Signed Numbers a. To add two numbers with the same signs, add their absolute values and attach the common sign. For example: (-3) + (-2) = -5 1.01 - 12 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide b. To add two numbers with opposite signs, find the difference of their absolute values, then attach the sign of the original number which had the greater absolute value. For example: (-2) + 3 = 1 c. Notice that -3 and +3 are the same distance but in opposite directions from 0 on the number line. What happens when you add two numbers like 3 and -3? 3 + (-3) = 0 −7 + 7 = 0 d. If the sum of two signed numbers is 0, the numbers are called additive inverses or opposites. For example: 7 - 3 = 4 is the same as: 7 + (-3) = 4 8 - 2 = 6 is the same as: 8 + (-2) = 6 e. It can be seen that subtracting a number is equivalent to adding its additive inverse or opposite. f. To subtract a signed number, add its opposite or additive inverse. In other words, change the subtraction symbol to addition and change the sign of the second signed number. For example: 5 - (-8) = 5 + (+8) « add +8 (Answer = 13) 6 - 11 = 6 + (-11) « add -11 (Answer = -5) −4 - (-7) = -4 + (+7) « add +7 (Answer = 3) 6. Multiplying and Dividing Signed Numbers a. The product of two numbers with like signs is a positive number. The product of two numbers with unlike signs is a negative number. In symbols: (+) × (+) = (+) (+) × (-) = (-) (-) × (-) = (+) (-) × (+) = (-) 1.01 - 13 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide b. Examples: (-4) × (-3) = (+12) (-4) × (+3) = (-12) c. The division of numbers with like signs gives a positive quotient. The division of numbers with unlike signs gives a negative quotient. In symbols: (+)/(+) = (+)

(+)/(-) = (-) (-)/(-) = (+) (-)/(+) = (-) d. Examples: (-24)/(-6) = (+4) (-24)/(+6) = (-4) e. Remember that multiplication is really a short form of addition. When we say +4 × (3), we are adding the number ­ 3 four times, that is, (-3) + (-3) + (-3) + (-3) = -12. Also, since division is a short form of subtraction, when we say ­ 24 ÷ (-6), we subtract the number -6 from the number -24 four times in order to reach 0, i.e., -24 - (-6) - (-6) - (-6) - (-6) = 0. Although we could repeat the process for the multiplication and division of any signed numbers, usage of the two rules will produce equivalent results. G. Exponents Objective 1.01.06 1. An exponent is a small number placed to the right and a little above another number, called the base, to show how many times the base is to be multiplied by itself. Thus, 34 (read "three to the fourth power") means 3 used as a factor four times or 3 × 3 × 3 × 3. In this case, 4 is the exponent, and 3 is the base. 2. In general, if b is any real number and n is any positive integer, the nth power of b is written bn (where b is the base and n is the exponent) and is read as "b to the nth power." This tells you that b is used as a factor n times. 3. Thus, 52 is called "5 raised to the second power" (or 5 "squared"), and 23 is called "2 raised to the third power" (or 2 "cubed"). When no exponent is shown for a number or no power is indicated, the exponent or power is understood to be 1. 1.01 - 14 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide Thus, 7 is the same as 71. Any number raised to the power of zero equals one; e.g., 70 = 1. Normally, exponents of zero and one are not left as the final value, but are changed to the simpler form of the base. 4. Exponents can be expressed as integers, as in the examples above, or as fractions or decimals such as 91/2 or 103.2 . They may also be positive or negative. 5. Exponents and powers are particularly useful in mathematics not only because they shorten the writing of mathematical expressions, but also because they simplify many mathematical operations. However, there are several special rules which govern mathematical operations involving numbers with exponents. 6. Addition and Subtraction a. The addition or subtraction of numbers with exponents can be performed only if both the bases and the exponents of the numbers are the same. When the bases of the exponents are different, the multiplication indicated by the exponent must be performed and the numbers then added or subtracted. Thus, 25 and 24 cannot be added directly because their exponents are different. They can be added only by carrying out the indicated multiplication first. Thus, 25 equals 2 × 2 × 2 × 2 × 2 which equals 32, and 24 equals 2 × 2 × 2 × 2 which equals 16. Therefore, 25 + 24 equals 32 + 16, which equals 48. b. When the bases and the exponents are the same, the numbers can be added or subtracted directly. For example: 35 + 35 = 2(35) = 2(243) = 486 7. Multiplication a. The multiplication of numbers with exponents of the same base is performed by adding the exponents. The general form is as follows: (am)(an) = a(m+n) b. It is important to remember that the bases of the numbers must be the same before they can be multiplied by adding their exponents. The base of the product is the same as the base of the two factors. Thus, 32 × 33 = 3(2+3) = 35 = 243 1.01 - 15 DOE-HDBK-1122-2009

Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 8. Division a. The division of numbers with exponents of the same base is performed by subtracting the exponent of the divisor (denominator) from the exponent of the dividend (numerator). The general form is: −am = a(m n) an b. Again, it is important to remember that the bases of the numbers must be the same before they can be divided by subtracting their exponents. The base of the quotient is the same as the base of the number divided. Thus, 52 (5 2) 3 2 = 2 − = 2 = 8 2 c. Division of numbers with exponents can be used to show why any number raised to the power of zero equals one. We know that any fraction in which the numerator equals the denominator can be reduced to 1; e.g., 2/2 = 1. Similarly: 32 (3 3) 0 3 = 2 − = 2 = 1 2 9. Exponent Raised to a Power a. Raising a number with an exponent to a power is performed by multiplying the exponent by the power. The general form is: (am)n = amn b. The base of the answer is the same as the base of the number raised to the power. Thus: (52)3 = 5(2×3) = 56 = 15,625 10. Product Raised to a Power a. Raising a product of several numbers to a power is performed by raising each number to the power. The general form is as follows: nbn(ab)n = a 1.01 - 16 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide b. Example: [(2)(3)(4)]2 = (22)(32)(42) = (4)(9)(16) = 576 c. This same result can also be obtained like this: [(2)(3)(4)]2 = 242 = 576 11. Mixed Product and Exponents Raised to a Power a. The same rule can be used to raise a product of several numbers with exponents to a power. The general form looks like this: (AaBbCc)n = A(a×n)B(b×n)C(c×n) b. Example: [(2)4(3)3(4)2]2 = (24×2)(33×2)(42×2) =(28)(36)(44) = (256)(729)(256) 12. Fraction Raised to a Power a. Raising a fraction to a power is performed by raising both numerator and denominator to the power. It should be remembered that with a proper fraction (i.e., numerator is less than the denominator) the resulting number must be less than one. Also, the resulting number will be less than the value of the original fraction. Thus, (2/3)3 equals 23/33, which equals 8/27, which is less than one and less than the original fraction, 2/3. 13. Negative Exponents and Powers a. A negative exponent or power has a special meaning. Any number, except 0, with a negative exponent equals the reciprocal of the same number with the same positive exponent. For example: −2 1 16 = = 62 36 b. The same rules for addition, subtraction, multiplication, division, and raising to a power apply to negative exponents that apply to positive exponents. However, in adding, subtracting, or multiplying the exponents, the rules for signed numbers must also be observed. 1.01 - 17 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 14. Fractional Exponents a. Fractional exponents are used to represent roots (see next m section). The general form of a fractional exponent is a n , which reads "the nth root of am." For example, a½, means the square root of a1, or a. In other words, a½ = a . 15. Calculator Method a. To raise a number to a power using a scientific calculator, use the following steps: 1) Enter the number. (Fractions must first be converted to decimal form.) yx2) Press the key. 3) Enter the power. (Fractions must first be converted to decimal form.)

4) Press the = key. The number displayed will be the number entered in step 1 raised to the power entered in step 3. H. Square Roots 1. To square a number means to multiply the number by itself, i.e., raise it to the second power. (Consider that a square is two- dimensional.) For example, 2 squared is 4, since 2 × 2 = 4. The square of 3 is 9, 4 squared is 16, and so on. 2. Just as subtraction "undoes" addition and division "undoes" multiplication, squaring a number can be "undone" by finding the square root. The general definition is as follows: If a2 = b, then a is a square root of b. 3. Be careful not to confuse the terms square and square root. For example, if 52 = 25, this indicates that 25 is the square of 5, and 5 is the square root of 25. To be explicit, we say that it is a perfect square because 5 times itself is 25. 4. All perfect squares other than 0 have two square roots, one positive and one negative. For example, because 72 = 49 and (-7)2 = 49, both 7 and -7 are square roots of 49. The symbol √, referred to as the radical, is used to write the principal, or positive, square root of a positive number. Objective 1.01.07 1.01 - 18 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 49 = 7 is read "The positive square root of 49 equals 7." 5. A negative square root is designated by the symbol − . 49− = −7 is read "The negative square root of 49 equals -7." 6. It is often convenient to use plus-or-minus notation: means the positive or negative square root of 49. 7. Therefore, the rule is that every positive real number a has two 49± square roots: a and − a . 8. It follows from the definition of square root that ( a )2 = a ,and that a2 = a . Because the square of every real number is either positive or zero, negative numbers do not have square roots in the set of real numbers. 9. Notice that ⋅ =4 25 100 =10 , and 4 ⋅ 25 2 5 .= ⋅ = 10 Therefore, in general, we can say: a. For any non-negative real numbers a and b: a b a b⋅ = ⋅ a a b. It also follows that = b b 10. Calculator Method a. To calculate the square root of any number using a scientific calculator, follow these steps: 1) Enter the number. (Fractions must first be converted to decimal form.) 2) Press the key. The number displayed will be the x square root of the number entered in step 1. An yxalternate method is to press the key and then type 0.5. This raises the number in step 1 to the power of 0.5, or ½. 1.01 - 19 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 11. Other roots a. For informational purposes only, we mention the fact that other roots may be found for a number. One of these is the cube root. To cube a number means to multiply the number by itself three times, i.e., raise it to the third power. (Consider that a cube is three-dimensional.) For example, 2 cubed is 8, since 2 × 2 × 2 = 8. The cube root (or third root) of a number, then, is the number that, when raised to the third power (cubed), equals the first number. The notation for a cube root is 3 a . b. Note that any root may be taken from a number to "undo" an exponent, such as the fourth or fifth root. The general definition for a root is: If an = b, then a is the nth root of b. c. The notation for the nth root is n a . These roots follow the same general rules as the square root. I. Scientific Notation Objective 1.01.08

1. The difficulty in writing very large or very small numbers in the usual manner is that a large number of zeros are required to write these numbers. This difficulty is overcome by using scientific notation, in which integral powers of ten are used instead of a large number of zeros to indicate the position of the decimal point. In addition to simplifying the writing of very large or very small numbers, scientific notation clearly identifies the number of significant digits in a number and simplifies arithmetic calculations involving multiplication, division, or raising to a power. For these reasons, it is good practice to write numbers in scientific notation when these operations are involved. 2. Converting From Standard Form To Scientific Notation There are two steps involved in writing a number in scientific notation. a. b. Move the decimal point just to the right of the first significant digit. The first significant digit is the first non-zero digit counting from the left. Indicate multiplication of the resulting number by a power of ten that makes its value equal to the original value. The power of ten is found by counting the number of places the decimal point was moved from its original position. If counted to the left, the power is positive; if counted to the right, it is negative. For example: 1.01 - 20 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 1) Suppose you want to express a number such as 700 in scientific notation. 2700 7 10 ss = × 2) Suppose you want to express 0.0014 in scientific notation. 30.0014 1.4 10−= ×rrr 3. Converting from Scientific Notation to Standard Form a. To transform from scientific notation to standard form, follow the opposite procedure. 51.96 10 1.96000 196,000 × = rrrrr = 22.27 10 002.27 0.0227× − = = ss b. There are two parts of a number written in scientific notation, the significant digits and the power of ten. Thus, in the number 3.21 × 106, 3, 2, and 1 are the significant digits and 106 is the power of ten. c. The ability to clearly see the number of significant digits can be helpful in performing arithmetic calculations. For example, the number of significant digits which should be reported in the product of two numbers can be readily determined if the two numbers are first written in scientific notation. d. When numbers are expressed in scientific notation, calculations can be more easily visualized. This is because they involve only numbers between 1 and 10 and positive and negative integral powers of ten which can be treated separately in the calculations using the rules for numbers with exponents. 4. Addition and Subtraction Using Scientific Notation Objective 1.01.09 a. Addition and subtraction cannot normally be performed directly using scientific notation because they require adding or subtracting digits of equal place value. Thus, when numbers expressed in scientific notation are to be added or subtracted, they must first be converted to forms having equal place value. This is commonly done by expressing them as numbers which are multiplied by the same integral power of ten. 1.01 - 21 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide The sum or difference of these significant digits, multiplied by their common power of ten, is the sum or difference of the original numbers. b. For example: (3.54 × 105) + (2.51 × 104) 3.54 × 105 is first changed to 35.4 × 104 35.4×10 4

+35.4×10 4 37.91 104 = ×105× 3.79 5. Multiplication and Division Using Scientific Notation a. Multiplication or division of numbers using scientific notation is performed by multiplying or dividing the significant digits and the powers of ten separately. The significant digits are multiplied or divided in the same manner as other mixed decimals. The powers of ten are multiplied or divided by adding or subtracting their exponents using the rules for multiplication and division of numbers with exponents. For example: (2.7 × 102)(3.1 × 10-3) = (2.7)(3.1) × (102)(10-3) = 8.37 × 10-1 which should be rounded off to 8.4 × 10-1. b. One of the most useful applications of scientific notation is in arithmetic calculations which involve a series of multiplications and divisions. The use of scientific notation permits accurate location of the decimal point in the final answer. c. Example: Perform the following calculation using scientific notation: (219)( 0.00204) (21.2)( 0.0312 ) 1) Write each term in scientific notation: 2 −3(2.19×10 )( 2.04×10 ) 1 −2(2.12×10 )( 3.12×10 ) 1.01 - 22 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 2) Multiply and divide the significant digits: ( )( ) ( )( ) 2.19 2.04 4.46 0.675 2.12 3.12 6.61 = = 3) Multiply and divide the powers of ten by adding and subtracting exponents: ( )( ) ( )( ) 2 3 1 1 1 0 11 2 10 10 10 10 10 1010 10 − − − + −− = = = 4) Combine the results: 0.675 × 100 = 0.675 = 6.75 × 10-1 6. "E" Notation a. An alternate method for annotating scientific notation is often used by pocket calculators, computers, and some references. The method uses an E in place of the "× 10," and the number written after the E is the exponent of 10. The standard and alternate methods for scientific notation are equivalent and can be converted from one form to another without a change in value. The examples below use both methods in equivalent expressions: 3.79 ×105 = 3.79E5 4.02 ×10-6 = 4.02E-6 5.89 ×100 = 5.89E0 7. Using "E" Notation with a Calculator a. Numbers in scientific notation are entered into a scientific calculator as follows: 1) Enter the significant digits. 2) Press the vary.) E or EXP key. (Actual key label may 3) Enter the power of 10. If the power is negative press the +/! key in conjunction with entering the power. J. Order of Mathematical Operations Objective 1.01.010 1.01 - 23 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 1. In solving any equation it is necessary to perform the operations in the equation in accordance with a certain hierarchy or order of operations. Equations are solved by simplifying operations of higher order first, according to group, left to right. The order for solving equations is as follows: a. Simplify expressions within grouping symbols, beginning with the innermost set if more than one set is used. b. Simplify all powers. c. Perform all multiplications and divisions in order from left to right. d. Perform all additions and subtractions in order from left to right. 2. For example: (3 + 1)2 × 3 ! 14 ÷ 2 a. Simplify parentheses. ( )3 1 2 3 2+ 14 × − ÷ { b. Simplify powers. ( )4 2 3 14 2× − ÷ { c. Perform multiplication and division left to right. 16 3 14 2× − ÷ { { d. Perform subtraction. 48 − =7 41 (Final Answer) K. Algebra Objective 1.01.011 1. Algebra is the branch of mathematics which deals with the

manipulation of words and letters, generically called symbols, which represent numbers. Two factors contribute to the widespread use of algebra in scientific calculations. First, by using words and letters to represent the values of physical quantities, physical relationships can be expressed in clear, concise, and completely generalized form. 1.01 - 24 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide Second, by using words and letters in place of numbers, combinations of physical relationships may be simplified to yield results applicable to any set of numbers. 2. For example, the area of a rectangle equals the product of the length of the rectangle multiplied by its width. In generalized terms, this statement can be written as: Area = Length × Width 3. This expression is a simple rule which tells the relationship between the area and the length and width of a rectangle. It does not mean that words are multiplied together but rather that numbers are inserted for the length and the width to obtain the area. For example, if the length is 4 feet and the width is 2 feet, the area is 2 feet × 4 feet or 8 square feet. This expression can be further simplified by using symbols or letters instead of words. For example, if area is designated by the letter A, length designated by the letter l, and width designated by the letter w, the following expression results: A = l × w or A = lw 4. In algebraic expressions, when two or more letters representing numbers are written next to each other without a symbol between them, multiplication is indicated. 5. Variables vs. Numbers When words or letters are used to represent numbers, they are called variables. Thus, when letters like x, y, z, f, or k are used to represent the values of physical quantities, they are called variables because their value varies with the actual numbers they may be chosen to represent. In the area calculation above, A, l, and w are variables used to represent the numerical values of area, length and width, respectively. 6. Properties of Variables a. Recall that every number has a sign and an exponent associated with it. Recall also that any number can be written as a fraction by putting that number as the numerator and 1 as the denominator, e.g., 5 = 5/1. These properties also apply to any symbols that we might use to represent numbers. Additionally, a symbol by itself stands for one of whatever the variable represents. That is to say, the symbol a by itself means "one of the variable represented by the letter a", or 1a. 1.01 - 25 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide Combining this with the other "invisible" properties mentioned, the symbol a is understood to represent "positive one of the variable represented by the letter a to the power of one, over 1," which would be expressed as: 1 1+ a = a 1 b. An expression that is either a numeral, a variable or the product of a numeral and one or more variables is called a monomial. A combination or sum of monomials is called a polynomial. Examples of each are: 1) Monomials 12 z 2 3 r −4x3 2) Polynomials 3x + 9 6a2 - 15 7. Equations a. An equation is a statement that indicates how two quantities or expressions are equal. The two quantities are written with an equal sign (=) between them. For example, 1 + 1 = 2 10 = 6 - (-4) 5 × 3 = 15 18 ÷ 2 = 9

are all equations because in each case the quantity on the left side is equal to the quantity on the right side. b. In algebra we use variables to represent numbers in equations. In this lesson we will manipulate and solve equations involving more than one variable, but we will find the solution, i.e., the final answer, to equations having only one variable. 8. Algebraic Manipulation a. The basic principle, or axiom, used in solving any equation is: whatever operation is performed on one side of an equation, be it addition, subtraction, multiplication, division, raising to an exponent, taking a root, must also be performed on the other side if the equation is to remain true. This principle must be adhered to in solving all types of equations. 1.01 - 26 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide b. This axiom can be thought of by visualizing the balancing of a scale. If the scale is initially balanced, it will remain balanced if the same weight is added to both sides, if the same weight is removed from both sides, if the weights on both sides are increased by the same factor, or if the weights on both sides are decreased by the same factor. c. Here are the general forms for algebraic manipulation of equations. For the real numbers a, b, c and n: See Table 2. - "Rules for Algebraic Manipulation" 1) Addition If a = b, then a + c = b + c 2) Subtraction If a = b, then a - c = b - c 3) Multiplication If a = b, then a x c = b x c 4) Division If a = b, then a ÷ c = b ÷ c 5) Involution If a = b, then an = bn Involution - raising to an exponent. 6) Evolution Evolution - taking a root. If a = b, then n a = n b 9. Manipulating and Solving Linear Equations a. The addition or subtraction of the same quantity from both sides of an equation may be accomplished by transposing a quantity from one side of the equation to the other. Transposing is a shortened way of applying the addition or subtraction axioms. Any term may be transposed or transferred from one side of an equation to the other if its sign is changed. Thus, in the equation below the +4 can be transposed to the other side of the equation by changing its sign: 5x + 4 = 14 (5x + 4) - 4 = (14) - 4 5x = 14 - 4 5x = 10 1.01 - 27 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide b. c. Transposing also works with multiplication and division. Remembering that any number can be expressed as a fraction we can rewrite the last line of the equation above. We can then move the 5 in the numerator of the left side to the denominator of the right side: 5x 10 = 1 1 x 10 = 1 5 10 x = 5 x = 2 Some linear equations may contain multiple terms (monomials) involving the same variable. In order to simplify the equation like terms must be combined. Don't forget those "invisible properties of variables." Here's an example: Solve for x in the equation: 3x - 5 = x + 3 1) Subtract x from both sides: 3x - 5 - x = x + 3 - x 2) Combine like terms and cancel: 2x - 5 = 3 3) Add 5 to both sides: 2x - 5 + 5 = 3 + 5 4) -5 and +5 cancel: 2x = 8 5) Divide both sides by 2: 2x 8 = 2 2 6) 2 over 2 cancels. Reduce. x = 4 1.01 - 28 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 10. Quadratic Equations

a. In manipulating an equation involving multiple variables, the "variable of interest" (or the variable to be solved for) must be moved to one side of the equal sign and all other variables must be moved to the other side. In order to accomplish this, operations must be performed on both sides of the equation that will result in a variable, or group of variables, to be canceled out from one side. This cancellation can only occur if the "opposite function" is performed on a function that already exists on that side of the equation. This means that a variable that is being multiplied can be canceled by dividing by the same variable. Addition can be canceled with subtraction, multiplication with division, etc. b. Example: Solve for a in the equation: a + b = c 1) Subtract b from both sides: a + b - b = c - b 2) +b and -b cancels, leaving: a = c - b c. Example: Solve for a in the equation: ab = c 1) Divide both sides by b: ab c = b b 2) b over b cancels, leaving: c a = b 1.01 - 29 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide d. e. Do not forget that the order of operations must be observed when manipulating equations. Otherwise a completely different solution may result. The key is to do the opposite function in reverse order. Here is an example which shows how this is done. Solve for a in the equation: a b+ = d c 1) Multiply both sides by c: (a b)+ c ⋅ = d ⋅c c 2) c over c cancels: a b dc + = 3) Subtract b from both sides: a b b b+ − = dc − 4) b - b cancels, leaving: a dc −b= Once the order of the arithmetic functions has been established, manipulation of the formula can begin. In the example above, if the values for a, b, and c were known, the first step would be to add b to a. The second step would be to divide by c. Therefore, in order to solve it, we do the opposite functions in reverse order. So, we first multiply by c. Then, we would subtract b. It is a good idea to rewrite the equation each time so that the operations can be reevaluated before the next step. One final example: Solve for d in the equation: ab2 = cd 2 1) Divide both sides by c: ab2 cd 2 = c c 1.01 - 30 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 2) c over c cancels: ab2 2= d c 3) Take the square root of both sides: 2 2ab d c = 4) Square root of square cancels, leaving: ab2 = d c 11. Substitution Linear equations are solved by combining like terms and reducing to find the solution. Quadratic equations are solved by substituting given values into the equation for all but one of the variables, thus making it a linear equation. The best approach is to first solve for the variable of interest by algebraic manipulation. Then find the solution by substituting the given values into the equation for the respective variables. The single, unknown variable, or the variable of interest, will be left on one side, being set equal to the solution. For example: Given the equation: 2x - y2 = 3a - b Where x = 5, y = (-4) and a = 3; solve for b a. To solve for b: 2x - y2 = 3a - b b. Add b to both sides: 2x - y2 + b = 3a - b + b c.  b + b cancels: 2x - y2 + b = 3a d. Transpose 2x - y2 to right side: b = 3a - 2x + y2 1.01 - 31 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide e. Substitute known values: b = 3(3) - 2(5) + (-4)2 f. Perform operations: b = 9 - 10 + 16 g. Simplify:

b = 15 Objective 1.01.012 L. Logarithms 1. In many cases, arithmetic operations can be performed much more quickly if the numbers involved are numbers with exponents to the same base. For example, the multiplication or division of numbers with exponents to the same base can be performed by merely adding or subtracting the exponents. Raising to a power or taking a root can be performed by merely multiplying or dividing the exponents by the power or root. It is this feature of numbers with exponents which led to the development of logarithms. If all numbers could be readily written as numbers with exponents to the same base, multiplication, division, raising to powers and taking roots could be performed much more quickly. 2. Any number can be expressed as a power of any other number. For example, 64 equals 26, 43, or 82. 64 also equals 72.137 or 101.806 . The use of logarithms involves expressing numbers as powers of a common number, such as 10, so that arithmetic operations with these numbers can be performed more quickly. 3. Simply put, a logarithm is an exponent. More explicitly, the logarithm of a number (n) is the exponent (x) of a given base (B) that is required to produce that number. The symbol log is used to denote taking a logarithm. The base is usually indicated by a small number written to the right and slightly below the symbol log. The general relationship and form are as follows: If n = Bx; where B > 0 and B ≠ 1; then: logBn = x 4. For example: 1000 = 103 → log101000 = 3 1.01 - 32 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 5. 6. 7. 8. 9. This says that the base ten logarithm of 1000 is 3, which means that the base number, 10, must be raised to the power of 3 to equal 1000. Here are some additional examples: 23 = 8 → log2 8 = 3 43/2 = 8 → log4 8 = 3/2 Before the development of the pocket calculator, the use of logarithms saved considerable computation time. For example, the evaluation of the following expression by hand would take a very long time. 895 3 0.0247 2 93,800 (0.00186)( 4.53)2 ( ) ( ) 1 ( ) However, using logarithms the above expression could be evaluated in a matter of minutes. Thus, logarithms, or logs, became one of the most useful tools in mathematics. In addition to simplifying arithmetic calculations and shortening computation time, logs are also important in engineering applications. The relationship between a number and its logarithm is used frequently to assist in measuring physical quantities when they vary over a wide range. For example, logarithmic scales are used to measure the neutron flux in nuclear reactors. Logarithms are also used for scales on charts and meters. Properties of Logarithms a. Since logarithms are exponents, the basic rules of exponents can be used to develop several useful properties of logarithms. Suppose that a, x, and y are numbers, and a is a suitable base for a logarithm (a > 0, a ≠ 1). The product rule for exponents says: y (x + y) ax ≅ a = a b. Let us say that: u = ax and v = ay 1.01 - 33 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide c. If we write each of these in logarithmic form we would have: x = loga u and y = loga v d. Then: u · v = ax · ay = a(x + y) e. If we write this in logarithmic form it would be: loga (u · v) = x + y f. If we substitute the values for x and y from above we have: g. loga (u · v) = loga u + loga v

This results in one of the rules for logarithms, the product rule. Using similar methods, we could also prove the other two rules that have been developed for logarithms. See Table 3 - "Rules for Logarithms" 10. Base Ten Logarithms a. b. c. Logs with the base of 10 are the most commonly used logarithms because of their relationship to the place values in the decimal system. Because of their wide use, base ten logarithms are often referred to as common logarithms. Observe the patterns in the number line and table. Notice the relationship between the power of ten and the logarithm. Any number can be expressed as a power of ten. Thus, 10 equals 101, 1,000 equals 103, 64 equals 101.806 and 527.3 equals 102.722 . Once a number has been expressed as a power of ten, the base ten logarithm of the number is known it is the exponent of 10. Thus log10 10 equals 1, log10 1000 equals 3, log10 64 equals 1.806 and log10 527.3 equals 2.722. Since base ten logarithms are so commonly used, the subscript 10 is often omitted after the symbol log. Thus, log 27.3 means the logarithm of 27.3 to the base 10. Refer students to number of line showing powers of 10 and Table 4 - "Log to Power of 10 Relationship." d. A common logarithm is most often a mixed number consisting of a whole number part and a decimal fraction part. The whole number part is called the characteristic of the logarithm. The decimal fraction part is called the mantissa. For example, in the logarithm of 527.3, which equals 2.722, the characteristic is 2 and the mantissa is 0.722. The mantissas of most logarithms are rounded off to a specified number of significant digits. Typically, mantissas are rounded off to three, four, five or more significant digits. 1.01 - 34 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 11. Log-Table Method a. To find the common logarithm of a number using four-place log tables, use the following steps: See Table 5 - "Four­ place Logarithms" 1) Write the number in scientific notation with up to four significant digits. 2) Using the product rule for logarithms, write the log of the product as the sum of the logs of the factors. 3) Determine the mantissa as follows: a) Find the row in the table corresponding to the first two digits of the number, then move over to the column corresponding to the third digit of the number. b) Find the number under the proportional parts section corresponding to the fourth digit of the number, and add it to the last digit of the decimal obtained in step 3.a. 4) Determine the characteristic by using the power of 10 written in step 1 and write the logs in the form of a sum. NOTE: If the power of 10 is negative, the log may be left in this form. NOTE: If the power of 10 is negative, the mantissa will be changed because of the subtraction of a whole number. Do examples in the study guide. b. As you may have observed, the mantissa of the base ten logarithm of a number depends only on the succession of significant digits in the number. The position of the decimal point in the number does not affect the mantissa. Of course, the characteristics are different for each of these numbers. See Table 6 - "Mantissa for Successive Significant Digits." c. Note that any time the logarithm of a number is rounded off it would be considered an approximate answer since each digit is necessary to exactly duplicate the number when the base is raised to that exponent. Since each significant digit of the logarithm affects the actual value of the number, a standard of four significant digits should maintained to ensure appropriate accuracy in the answers.

12. Calculator Method 1.01 - 35 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide Since hand-held scientific calculators are readily available today, it is impractical to use log tables. To find the logarithm of a number with a calculator: a. Enter the number. b. Press the log key. The number displayed is the logarithm of the number entered in step 1. 13. Natural Logarithms a. A logarithm can be written to any base. For most practical computations, base ten logarithms are used because of their relationship to the place values in the decimal system. However, in many scientific and engineering problems, it is convenient to use another base, symbolized by the letter e. e is an irrational number whose value is 2.71828 . . . The actual value of e is the limiting value of (1 + 1/n)n as n gets larger and larger. b. Although it is an irrational number, it can still be used as the base for logarithms in the same way as 10 is used for base ten logarithms. e is the basis for many laws of nature, such as the laws of growth and decay of physical quantities, including the decay of radioactive substances and the growth and decay of neutron population in a nuclear reactor. Because of the relationship of e to natural phenomena, logarithms to the base e are called natural logarithms. c. The natural logarithm of a number is the exponent to which e must be raised in order to get that number. The symbol ln is used to denote a natural logarithm which is the same as saying loge. The relationship is expressed as follows: If ex = n then ln n = x d. For example: 1) ln 2 = 0.693147 0.693147... = 2. which means that e 2) ln 10 = 2.302585 2.302585... = 10. which means that e 3) ln e = 1 which means that e1 = e 1.01 - 36 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide e. Natural logarithms are not often used for computations. However, they appear frequently in decay and shielding calculations problems because of the relationship of e to natural phenomena. As a result, it is important to know how to determine the natural logarithms of numbers. f. Tables of natural logarithms are available in several standard handbooks. However, there are several important differences between natural logarithms and base ten logarithms which must be understood to use natural logarithms. A natural logarithm is not separated into a characteristic and a mantissa. This is because the whole number part of a natural logarithm does not relate to the position of the decimal point. Therefore, tables of natural logarithms give the entire logarithm, not just the decimal fraction part. Moreover, if a natural logarithm is negative, the entire logarithm is negative and is shown as such in a table of natural logarithms. Further, there is no part of the natural logarithm of a number which is not affected by the position of the decimal point. For all these reasons, tables of natural logarithms cannot be made concise. g. To find the natural log of a number using a hand-held calculator: 1) Enter the number. 2) Press the ln key. The number displayed is the natural logarithm of the number entered in step 1. 14. Antilogarithms

a. An antilogarithm, usually shortened to "antilog," is the opposite of a logarithm and is much easier to do. The antilog of a given number is the value obtained by raising the base to that number. Finding antilogs is an important part in the overall use of logarithms in computations. If numbers are converted to logarithms to perform calculations, the answer must be converted back from logarithms once the calculations have been performed. The symbol log-1 is used in calculations to indicate the antilog is going to be taken. The base of 10 is assumed unless otherwise noted. The general form is: log-1 x = n which means 10x = n b. For example: log-1 3 which means 103 = 1000 Refer also to example in study guide. 1.01 - 37 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide c. On a scientific calculator the base 10 antilog of a number is obtained by raising the base (10) to that number. 1) Enter the number. See Table 7 -log−1 or 10x key. The number 2) Press the "Summary of Log displayed is the antilog of the number of entered in Definitions." step 1. In other words, 10 raised to that power. d. The symbol ln-1 is used to denote the inverse natural log, i.e. the antilog of base e. ln-1 x = n which means ex = n e. For example: 0.693... = 2 ln-1 0.693 which means e f. On a scientific calculator the inverse natural log of a number is obtained as follows: 1) Enter the number. ex2) Press the Ln-1 or key. The number displayed is the inverse natural log of the number of entered in step 1. In other words, e raised to that power. 14. Solving for Variables as Exponents a. One of the useful applications for logarithms is to solve algebraic equations with unknown exponents. In the following example, for instance, if the exponent is not known, it would be difficult to determine the correct value of x in order to make the statement (or equation) true. 2356 = 3x b. With the use of logarithms, however, this type of problem can be easily solved. The steps for solving an equation of this type are: 1) Make sure the base raised to the unknown exponent is isolated on one side of the equation (this may involve some manipulation of the formula in more complicated equations). 2) Take the log of both sides of the equation: log 2356 = log 3x 1.01 - 38 DOE-HDBK-1122-2009 Module 1.01 Basic Mathematics and Algebra Instructor’s Guide 3) Since a logarithm is an exponent, and this is an exponent raised to a power, this statement can now be rewritten using the laws of exponents, which say that the exponent and power are multiplied: log 2356 = (log 3)(x) 4) Divide both sides by log 3 which moves it to the right side of the equation: log 2356 (log 3 )( x) = log 3 log 3 5) Cancel terms and rewrite the equation: log 2356 = x log 3 6) Perform the operations and solve: 3.372 = x 0.477 7.068 = x c. This answer can now be checked by substituting it back into the original equation to see if it makes the statement true: 2356 = 37.068 d. Some problems may involve the base of the natural logarithm, e, raised to an unknown power. This exponent can be determined by isolating e on one side of the equation and then taking the natural log of both sides. This is done because taking the natural log of e reduces to 1. For example: n125 = 1000e 125 n= e 1000 125 nln = ln e 1000 ln 0.125 = (ln e)(n) ln 0.125 = (1)(n) -2.0794 = n 1.01 - 39 DOE-HDBK-1122-2009

Module 1.01 Basic Mathematics and Algebra Instructor’s Guide III. SUMMARY A. Review major topics 1. Fractions 2. Decimals 3. Percent 4. Signed Numbers 5. Exponents 6. Square Roots 7. Scientific Notation 8. Order of Mathematical Operations 9. Algebra 10. Logarithms B. Review learning objectives IV. EVALUATION Evaluation should consist of a written examination comprised of multiple choice questions. 80% should be the minimum passing criteria for the examination. 1.01 - 40 DOE-HDBK-1122-2009 Module 1.02 Unit Analysis and Conversion Instructor’s Guide Course Title: Radiological Control Technician Module Title: Unit Analysis & Conversion Module Number: 1.02 Objectives: 1.02.01 Identify the commonly used unit systems of measurement and the base units for mass, length, and time in each system. 1.02.02 Identify the values and abbreviations for SI prefixes. 1.02.03 Given a measurement and the appropriate conversion factor(s) or conversion factor table, convert the measurement to the specified units. 1.02.04 Using the formula provided, convert a given temperature measurement to specified units. References: 1. "Health Physics and Radiological Health Handbook"; Shleien; 1992. 2. DOE-HDBK-1010-92 (June 1992) "Classical Physics" DOE Fundamental Handbook; US Department of Energy. 3. "Chart of the Nuclides"; Sixteenth Edition, Knolls Atomic; 2003. Instructional Aids: 1. Overheads 2. Overhead projector/screen 3. Chalkboard/whiteboard 4. Lessons Learned 1.02 - 1 DOE-HDBK-1122-2009 Module 1.02 Unit Analysis and Conversion Instructor’s Guide I. MODULE INTRODUCTION A. Self Introduction 1. Name 2. Phone number 3. Background 4. Emergency procedure review B. Motivation A knowledge of the unit analysis and conversion process is a necessity for the RCT. It is useful for air and water sample activity calculations, contamination calculations, and many other applications. C. Overview of Lesson 1. Units 2. Unit systems 3. Unit analysis and conversions including use of conversion factor tables O.H.: Objectives D. Introduce Objectives II. LESSON OUTLINE A. Units and Measurements 1. Units are used to express physical quantities or measurements, i.e., length, mass, etc. All measurements are actually relative in the sense that they are comparisons with some standard unit of measurement. Two items are necessary to express these physical quantities: a. A number--expresses the magnitude. b. A unit--expresses the dimension. 2. A number and a unit must both be present to define a measurement. 3. Measurements are algebraic quantities and as such may be mathematically manipulated subject to algebraic rules. 1.02 - 2 DOE-HDBK-1122-2009 Module 1.02 Unit Analysis and Conversion Instructor’s Guide 4. Fundamental Quantities All measurements or physical quantities can be expressed in terms of three fundamental quantities. They are called fundamental quantities because they are dimensionally independent. They are: See Fig. 1 in the Study Guide - "Fundamental Units" a. Length (L) b. Mass (M) (not the same as weight) c. Time (T) 5. Derived Quantities Other quantities are derived from the fundamental quantities. These derived quantities are formed by multiplication and/or division of fundamental quantities. For example: a. Area is the product of length times length (width), which is L × L, or L2. b. Volume is area times length, which is length times length times length, or L3.

c. Velocity is expressed in length per unit time, or L/T. d. Density is expressed in mass per unit volume, or M/L3. B. Systems of Units Objective 1.02.01 1. The units by which physical quantities are measured are established in accordance with an agreed standard. Measurements made are thereby based on the original standard which the unit represents. The various units that are established, then, form a system by which all measurements can be made. 2. English System See Table 1 of the Study Guide - "English System Base Units" 1.02 - 3 DOE-HDBK-1122-2009 Module 1.02 Unit Analysis and Conversion Instructor’s Guide a. The system that has historically been used in the United States is the English System, sometimes called the English Engineering System (EES). Though no longer used in England, many of the units in this system have been used for centuries and were originally based on common objects or human body parts, such as the foot or yard. Though practical then, the standards for these units were variable as the standard varied from object to object, or from person to person. b. Even though fixed standards have since been established for these antiquated units, no uniform correlation exists between units established for the same quantity. For example, in measuring relatively small lengths there are inches, feet, and yards. There are twelve inches in a foot, and yet there are only three feet in a yard. This lack of uniformity makes conversion from one unit to another confusing as well as cumbersome. However, in the U.S., this system is still the primary system used in business and commerce. 3. International System of Units (SI) a. Since the exchange of scientific information is world-wide today, international committees have been set up to standardize the names and symbols for physical quantities. b. In 1960, the International System of Units (abbreviated SI from the French name Le Système Internationale d'Unites) was adopted by the 11th General Conference of Weights and Measures (CGPM). c. The SI, or modernized metric system, is based on the decimal (base 10) numbering system. First devised in France around the time of the French Revolution, the metric system has since been refined and expanded so as to establish a practical system of units of measurement suitable for adoption by all countries. The SI system consists of a set of specifically defined units and prefixes that serve as an internationally accepted system of measurement. Nearly all countries in the world use metric or SI units for business and commerce as well as for scientific applications. 4. SI Prefixes Objective 1.02.02 1.02 - 4 DOE-HDBK-1122-2009 Module 1.02 Unit Analysis and Conversion Instructor’s Guide See Table 2 of the a. The SI system is completely decimalized and uses prefixes for the base units of meter (m) and gram (g), as well as for derived units, such as the liter (l) which equals 1000 cm3. Study Guide - "SI Prefixes" b. SI prefixes are used with units for various magnitudes associated with the measurement being made. Units with a prefix whose value is a positive power of ten are called multiples. Units with a prefix whose value is a negative power of ten are called submultiples.

c. For example, try using a yard stick to measure the size of a frame on film for a camera. Instead you would use inches, because it is a more suitable unit. With the metric system, in order to measure tiny lengths, such as film size, the prefix milli- can be attached to the meter unit to make a millimeter, or 1/1000 of a meter. A millimeter is much smaller and is ideal in this situation. On the other hand, we would use a prefix like kilo- for measuring distances traveled in a car. A kilometer would be more suited for these large distances than the meter. See Table 3 in the d. Prior to the adoption of the SI system, two groups of units were commonly used for the quantities length, mass, and time: MKS and CGS. Study Guide - "Metric Subsystems" 5. SI Units a. There are seven fundamental physical quantities in the SI system. These are length, mass, time, temperature, electric charge, luminous intensity, and molecular quantity (or amount of substance). In the SI system there is only one SI unit for each physical quantity. The SI system base units are those in the metric MKS system. Table 4 in the Study Guide lists the seven fundamental quantities and their associated SI unit. The units for these seven fundamental quantities provide the base from which the units for other physical quantities are derived. See Table 4 in the Study Guide - "International System (SI) Units" b. For most applications the RCT will only be concerned with the first four quantities as well as the quantities derived from them. 6. Radiological Units In the SI system, there are derived units for quantities used for radiological control. These are the becquerel, the gray, 1.02 - 5 DOE-HDBK-1122-2009 Module 1.02 Unit Analysis and Conversion Instructor’s Guide and the sievert. The SI unit of activity is the becquerel, which is the activity of a radionuclide decaying at the rate of one spontaneous nuclear transition per second. The gray is the unit of absorbed dose, which is the energy per unit mass imparted to matter by ionizing radiation, with the units of one joule per kilogram. The unit for equivalent dose is the sievert, which has units of joule per kilogram. These quantities and their applications will be discussed in detail in Lesson 1.06. 7. Other units There are several other SI derived units that are not listed in Table 4 of the Study Guide. It should be noted that the SI system is evolving and that there will be changes from time to time. The standards for some fundamental units have changed in recent years and may change again as technology improves our ability to measure even more accurately. C. Unit Analysis and Conversion Process 1. Units and the Rules of Algebra a. Remember that a measurement consists of a number and a unit. When working problems with measurements, it should be noted that the measurement units are subject to the same algebraic rules as the values. Some examples are provided below. 2(cm) × (cm ) = cm 3ft 2= ft ft 1 = yr−1 yr b. As a result, measurements can be multiplied, divided, etc., in order to convert to a different system of units. Obviously, in order to do this, the units must be the same. For example, a square measures one foot in length and 18 inches in width. To find the area of the square in square inches we must multiply the length by the width. However, when the measurements are in different units, and cannot be multiplied directly. Objective 1.02.03 1.02 - 6 DOE-HDBK-1122-2009

Module 1.02 Unit Analysis and Conversion Instructor’s Guide c. We can convert feet to inches. We know that there are 12 inches in one foot. We can use this ratio to convert 1 foot to 12 inches. Then we can then calculate the area as 12 inches × 18 inches, which equals 216 in2, which is a valid measurement. 2. Steps for Unit Analysis and Conversion a. Determine given units and desired units. b. Build (or obtain) conversion factor(s) -- see Conversion Tables at end of lesson. 1) A conversion factor is a ratio of two equivalent physical quantities expressed in different units. When expressed as a fraction, the value of all conversion factors is 1. Because a conversion factor equals 1, it does not matter which value is placed in the numerator or denominator of the fraction. 2) Examples of conversion factors are: 365 days 1year 12inches 1 foot 1foot3 2.832E4cm 3 3) Building conversion factors involving the metric prefixes for the same unit can be tricky. This involves the conversion of a base unit to, or from, a subunit or superunit. 4) To do this, use the following steps: Example: 1 gram to milligrams a) Place the base unit in the numerator and the subunit/ superunit in the denominator (or vice versa): 1.02 - 7 DOE-HDBK-1122-2009 Module 1.02 Unit Analysis and Conversion Instructor’s Guide g mg b) Place a 1 in front of the subunit/superunit: g 1mg c) Place the value of the prefix on the subunit/ superunit in front of the base unit: m (milli-) = 10-3 or 1E-3 1E − 3g 1mg 5) Also remember that algebraic manipulation can be used when working with metric prefixes and bases. For example, 1 centimeter = 10-2 meters. This means that 1 meter = 1/10-2 centimeters, or 100 cm. Therefore, the two conversion factors below are equal: 1E − 2m 1m = 1cm 100cm c. Set up an equation by multiplying the given units by the conversion factor(s) to obtain desired unit(s). 1) When a measurement is multiplied by a conversion factor, the unit(s) (and probably the magnitude) will change; however, the actual measurement itself does not change. For example, 1 ft and 12 inches are still the same length; only different units are used to express the measurement. 2) By using a "ladder" or "train tracks," a series of conversions can be accomplished in order to get to the desired unit(s). By properly arranging the numerator and denominator of the conversion factor(s), given and intermediate units will cancel out by multiplication or division, leaving the desired units. Some examples of the unit analysis and conversion process follow: 1.02 - 8 DOE-HDBK-1122-2009 Module 1.02 Unit Analysis and Conversion Instructor’s Guide 3. Example 1: Convert 3 years to seconds. Step 1 - Determine given and desired unit(s): Given units: years Desired units: seconds. Step 2 - Build/obtain conversion factor(s): We can use multiple conversion factors to accomplish this problem: 1 year = 365.25 days 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds; 1 hour = 3600 seconds Step 3 - Analyze and cancel given and intermediate units. Perform multiplication and division of numbers: ⎜ ⎛ ⎜ ⎝ 3 years ⎟⎜ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 365.25days year ⎛ ⎜ ⎝ ⎞ ⎟⎟⎜ ⎠ 24hours ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎟ 3600sec⎞ ⎟ ⎠ =94,672,800sec day hour 4. Example 2: What is the activity of a solution in μ Ci if ml dpm it has 2000 ? gallon Step 1 - Determine given and desired unit(s): dpm Given units: gallon Desired units: μ Ci ml 1.02 - 9 DOE-HDBK-1122-2009

Module 1.02 Unit Analysis and Conversion Instructor’s Guide Step 2 - Build conversion factor(s): 1 liter = 0.26418 gallons 1 dpm = 4.5 E-07 µCi 1 liter = 1000 ml Step 3 - Analyze and cancel given and intermediate units. Perform multiplication and division of numbers. ⎛ 2000dpm ⎞⎛ 4.5E − 7μCi ⎞⎛ 0.26418gal ⎞⎛ 1l ⎞ μCi = 2.38E − 7⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ gal ⎠⎝ 1dpm ⎠⎝ 1l ⎠⎝ 1,000 ml ⎠ ml D. Temperature Measurements and Conversions 1. Temperature measurements are made to determine the amount of heat flow in an environment. To measure temperature it is necessary to establish relative scales of comparison. Three temperature scales are in common use today. The general temperature measurements we use on a day-to-day basis in the United States are based on the Fahrenheit scale. In science, the Celsius scale and the Kelvin scale are used. 2. The Fahrenheit scale, named for its developer, was devised in the early 1700's. This scale was originally based on the temperatures of human blood and salt-water, and later on the freezing and boiling points of water. Today, the Fahrenheit scale is a secondary scale defined with reference to the other two scientific scales. The symbol °F is used to represent a degree on the Fahrenheit scale. 3. About thirty years after the Fahrenheit scale was developed, Anders Celsius, a Swedish astronomer, suggested that it would be simpler to use a temperature scale divided into one hundred degrees between the freezing and boiling points of water. For many years his scale was called the centigrade scale. In 1948 an international conference of scientists re-named it the Celsius scale in honor of its inventor. The Celsius degree, °C, was defined as 1/100 of the temperature difference between the freezing point and boiling point of water. Objective 1.02.04 See Fig. 2 in the Study Guide - "Comparison of Kelvin, Celsius and Fahrenheit scales" 1.02 - 10 DOE-HDBK-1122-2009 Module 1.02 Unit Analysis and Conversion Instructor’s Guide 4. In the 19th century, an English scientist, Lord Kelvin, established a more fundamental temperature scale that used the lowest possible temperature as a reference point for the beginning of the scale. The lowest possible temperature, sometimes called absolute zero, was established as 0 K (zero Kelvin). This temperature is 273.15°C below zero, or -273.15°C. Accordingly, the Kelvin degree, K, was chosen to be the same as a Celsius degree so that there would be a simple relationship between the two scales. 5. Note that the degree sign (°) is not used when stating a temperature on the Kelvin scale. Temperature is stated simply as Kelvin (K). The Kelvin was adopted by the 10th Conference of Weights and Measures in 1954, and is the SI unit of thermodynamic temperature. Note that the degree Celsius (°C) is the SI unit for expressing Celsius temperature and temperature intervals. The temperature interval one degree Celsius equals one kelvin exactly. Thus, 0°C = 273.15 K by definition. 6. To convert from one unit system to another, the following formulas are used: o o ( F −32) 1) C = 1.8 or ⎛ ⎞o o 5C = ( F −32)⎜ ⎟9⎝ ⎠ 2) F =1.8( )o oC + 32 or o ⎛ ⎞9 oCF = ( ) + 32⎜ ⎟5⎝ ⎠ 3) K = oC + 273.15 See Table 5 in the Study Guide - "Equations for Temperature Conversions" 1.02 - 11 DOE-HDBK-1122-2009 Module 1.02 Unit Analysis and Conversion Instructor’s Guide 7. Example 3: Convert 65° Fahrenheit to Celsius: o(65 F − 32)oC = 1.8 33 oC = 1.8 oC = 18.3oC III. SUMMARY

A. Review major topics 1. Units 2. Unit systems 3. Unit analysis and conversions including use of conversion factor tables B. Review learning objectives IV. EVALUATION Evaluation should consist of a written examination comprised of multiple choice questions. 80% should be the minimum passing criteria for the examination. 1.02 - 12 DOE-HDBK-1122-2009 Module 1.03 Physical Sciences Instructor’s Guide Course Title: Radiological Control Technician Module Title: Physical Sciences Module Number: 1.03 Objectives: 1.03.01 Define the following terms as they relate to physics: a. Work b. Force c. Energy 1.03.02 Identify and describe four forms of energy. 1.03.03 State the Law of Conservation of Energy. 1.03.04 Distinguish between a solid, a liquid, and a gas in terms of shape and volume. 1.03.05 Identify the basic structure of the atom, including the characteristics of subatomic particles. 1.03.06 Define the following terms: a. Atomic number b. Mass number c. Atomic mass d. Atomic weight 1.03.07 Identify what each symbol represents in the X notation. 1.03.08 State the mode of arrangement of the elements in the Periodic Table. 1.03.09 Identify periods and groups in the Periodic Table in terms of their layout. 1.03.10 Define the terms as they relate to atomic structure: a. Valence shell b. Valence electron References: 1. "Chart of the Nuclides"; Sixteenth Edition, Knolls Atomic; 2003. 2. "Modern Physics"; Holt, Rinehart and Winston, Publishers; 1976. 3. "Chemistry: An Investigative Approach"; Houghton Mifflin Co., Boston; 1976. 4. "Chemical Principles with Qualitative Analysis"; Sixth ed.; Saunders College Pub.; 1986. 5. "Introduction to Chemistry" sixth ed., Dickson, T. R., John Wiley & Sons, Inc.; 1991. 6. "Matter"; Lapp, Ralph E., Life Science Library, Time Life Books; 1965. 7. "Physics"; Giancoli, Douglas C., second ed., Prentice Hall, Inc.; 1985. 8. DOE/HDBK-1015 "Chemistry: Volume 1 of 2"; DOE Fundamentals Handbook Series; January 1993. 1.03 - 1 DOE-HDBK-1122-2009 Module 1.03 Physical Sciences Instructor’s Guide Instructional Aids: 1. Overheads 2. Overhead projector/screen 3. Chalkboard/whiteboard 4. Chart of the Nuclides 5. Periodic Table of the Elements 6. Lessons Learned I. MODULE INTRODUCTION A. Self-Introduction 1. Name 2. Phone number 3. Background 4. Emergency procedure review B. Motivation It is important to a RCT that they have a basic understanding of physics because they may work in an environments where materials can undergo changes in state, resulting in changes in the work environment. C. Overview of Lesson 1. Physics definitions 2. Law of Conservation of Energy 3. The Atom 4. Periodic Table 5. Valence Electrons O.H.: Objectives D. Introduce Objectives II. MODULE OUTLINE A. Work and Energy Physics is the branch of science that describes the properties, changes, and interactions of energy and matter. This unit will serve as a brief introduction to some of the concepts of physics as they apply to the situations that may be encountered by RCTs. A general definition of matter is anything that has mass and occupies space. Energy can be understood by relating it to another physical concept - work. 1.03 - 2 DOE-HDBK-1122-2009 Module 1.03 Physical Sciences Instructor’s Guide 1. Work and Force Objective 1.03.01 a. Work is defined in physics as a force acting through a distance. b. A force is a push or a pull. A more technical definition of force is

any action on an object that causes the object to change speed or direction. c. Units (1) Force is derived as the product of mass and acceleration. (2) The SI derived unit is expressed in terms of newtons, (N) Kg ×m N = 2S (3) Mathematically, work is expressed as: W = F x d where: W = Work F = Force (newtons) d = Distance (meters) (4) The SI unit of work is the joule (5) One joule of work is performed when a force of one newton is exerted through a distance of one meter. (6) Thus: J N m= × B. Energy Objective 1.03.02 1. Energy is defined as the ability to do work. 1.03 - 3 DOE-HDBK-1122-2009 Module 1.03 Physical Sciences Instructor’s Guide a. Kinetic energy describes the energy of motion an object possesses. For example, a moving airplane possesses kinetic energy. 1 2E = mvK 2 Where: m = mass v = velocity b. Potential energy indicates how much energy is stored as a result of the position or the configuration of an object. For example, water at the top of a waterfall possesses potential energy. E = mghp Where: m = mass g = free fall acceleration h = vertical distance c. Thermal energy describes the energy that results from the random motion of molecules. For example, steam possesses heat energy. d. Chemical energy describes the energy that is derived from atomic and molecular interactions in which new substances are produced. For example, the substances in a dry cell provide energy when they react. 2. Law of Conservation of Energy Objective 1.03.03 a. The Law of Conservation of Energy states that the total amount of energy in a closed system remains unchanged. Stated in other terms, as long as no energy enters or leaves the system, the amount of energy in the system will always be the same, although it can be converted from one form to another. 1.03 - 4 DOE-HDBK-1122-2009 Module 1.03 Physical Sciences Instructor’s Guide b. Gasoline contains chemical energy that is released in the form of heat when a chemical reaction (burning) with oxygen occurs. This energy comes from the breaking and making of bonds between atoms. New products, carbon dioxide and water, are formed as the gasoline combines with oxygen. The energy of the burning gasoline produces heat energy which causes the gaseous combustion products to do work on the pistons in the engine. The work results in the vehicle moving, giving it kinetic energy. c. Units (1) Thermal energy is often measured in units of calories (CGS) or British Thermal Units or BTUs (English). (a) A calorie is the amount of heat needed to raise the temperature of 1 gram of water by 1 °C. One calorie is equal to 4.18605 joules. (b) A BTU is the amount of heat needed to raise the temperature of 1 pound of water by 1 °F. One BTU is equal to 1.055E3 joules. (2) Electrical energy is sometimes expressed in units of kilowatt-hours. One kw-hr is equal to 3.6E6 joules (a) A very small unit used to describe the energy of atomic and subatomic size particles is the electron volt (eV). One electron volt is the amount of energy acquired by an electron when it moves through a potential of one volt. (b) It takes about 15.8 eV of energy to remove an electron from an atom of argon. (c) Superunits such as kiloelectron volt (keV) and megaelectron volt (MeV) are used to indicate the energies of various ionizing radiations. d. Work-Energy Relationship

(1) When work is done by a system or object, it expends energy. For example, when gaseous combustion products push against the pistons, the gas loses energy. The chemical energy stored in the gasoline is used to do work so that the car will move. 1.03 - 5 DOE-HDBK-1122-2009 Module 1.03 Physical Sciences Instructor’s Guide (2) When work is done on a system or object, it acquires energy. (3) The work done on the car by the combustion of the gasoline causes the car to move, giving it more kinetic energy. (4) When energy is converted to work or changed into another form of energy, the total amount of energy remains constant. Although it may appear that an energy loss has occurred, all of the original energy can be accounted for. (5) Consider the example of the automobile. The energy stored in the gasoline is converted to heat energy, some of which is eventually converted to kinetic energy. The remainder of the heat energy is removed by the engine's cooling system. The motion of the engine parts creates friction, heat energy, which is also removed by the engine's cooling system. As the car travels, it encounters resistance with the air. If no acceleration occurs, the car will slow down and the kinetic energy is converted to friction or heat energy. The contact of the tires on the road converts some of the available kinetic energy to heat energy (friction), slowing down the car. A significant amount of the energy stored in the gasoline is dissipated as wasted heat energy. e. Energy-mass relationship Energy can also be converted into mass and mass converted into energy. This topic will be discussed further in Section 1.04 Nuclear Physics. C. Energy and Change of State 1. Matter is anything that has mass and takes up space. 2. There are three states of matter solid, liquid and gas. See Fig. 1 - "Energy Conversion in an Automobile" Objective 1.03.04 See Table 1 - "State of Matter Compared" and Fig. 2 - "States of Matter" 1.03 - 6 DOE-HDBK-1122-2009 Module 1.03 Physical Sciences Instructor’s Guide 3. Solid State a. A solid has definite shape and volume. The solid state differs from the liquid and gaseous states in that: (1) The molecules or ions of a solid are held in place by strong attractive forces. (2) The molecules still have thermal energy, but the energy is not sufficient to overcome the attractive forces. (3) The molecules of a solid are arranged in an orderly, fixed pattern. 4. Liquid State (a) When heat is added to a substance, the molecules acquire more energy, which causes them to break free of their fixed crystalline arrangement. As a solid is heated, its temperature rises until the change of state from solid to liquid occurs. (b) The volume of a liquid is definite since the molecules are very close to each other, with almost no space in between. Consequently, liquids can undergo a negligible amount of compression. However, the attractive forces between the molecules are not strong enough to hold the liquid in a definite shape. For this reason a liquid takes the shape of its container. (c) High energy molecules near the surface of a liquid can overcome the attractive forces of other molecules. These molecules transfer from the liquid state to the gaseous state. If energy (heat) is removed from the liquid, the kinetic energy of the molecules decreases and the attractive forces can hold the molecules in fixed positions. When compared with the kinetic energy, the attractive forces are not strong enough to hold the molecules in fixed positions, forming a solid.

5. Gaseous State (a) If the temperature of a liquid is increased sufficiently, it boils, that is, molecules change to the gaseous state and escape from the surface. Eventually, all of the liquid will become a gas. 1.03 - 7 DOE-HDBK-1122-2009 Module 1.03 Physical Sciences Instructor’s Guide (b) A gas has both indefinite shape and indefinite volume. A large space exists between gas molecules because of their high thermal energy. This allows for even more compression of a substance in the gaseous state. D. The Atom Objective 1.03.05 1. The Bohr Model was described by Ernest Rutherford and Niels Bohr – 1911 See Fig. 3 - Atomic Model a. Made of protons, neutrons, and electrons b. Central core called the nucleus c. Contains protons and neutrons d. Nuclear forces hold nucleus together 2. Protons a. Positively charged (+1) b. Mass = 1.6726 x 10-24 gm or 1.007276470 amu c. Each element is determined by the number of protons in its nucleus. All atoms of the same element have the same number of protons. 3. Neutrons a. Neutrally charged (0) b. Mass = 1.6749 x 10-24 gm or 1.008665012 amu c. Determines the isotope of an element. Same number of protons (therefore, of the same element) but different number of neutrons. Does not affect chemical property of element. 4. Electrons a. Negatively charged (-1) b. Small mass = 9.1085 x 10-28 or 0.00054858026 amu (1/1840 of a proton) c. The mass of an electron is so small as compared to that of a proton or neutron, virtually the entire mass of an atom is furnished by the nucleus. 1.03 - 8 DOE-HDBK-1122-2009 Module 1.03 Physical Sciences Instructor’s Guide d. Number of electrons is normally equal to the number of protons (atom is electrically neutral) e. The number of electrons in the outermost shell determines the chemical behavior or properties of the atom. E. The Elements 1. Even though all atoms have the same basic structure, not all atoms are the same. There are over a hundred different types of atoms. These different types of atoms are known as elements. The atoms of a given element are alike but have different properties than the atoms of other elements. 2. Elements are the simplest forms of matter. They can exist alone or in various combinations. Different elements can chemically combine to form molecules or molecular compounds. For example, water is a compound, consisting of water molecules. These molecules can be decomposed into the elements hydrogen and oxygen. The elements hydrogen and oxygen are fundamental forms of matter. They cannot be further separated into simpler chemicals. 3. Chemical Names See Table 2 - "List of Elements by a. Currently, there are more than 110 named elements. Some have Name" been known for many centuries, while others have only been discovered in the last 15 or 20 years. Each element has a unique name. The names of the elements have a variety of origins. Some elements were named for their color or other physical characteristics. Others were named after persons, places, planets or mythological figures. b. For example, the name chromium comes from the Greek word chroma, which means "color." Chromium is found naturally in compounds used as pigments. The elements curium, einsteinium, and fermium were named after famous nuclear physicists. Germanium, polonium and americium, were named after countries. Uranium, neptunium and plutonium are named in sequence for the three celestial bodies Uranus, Neptune and

Pluto. 1.03 - 9 DOE-HDBK-1122-2009 Module 1.03 Physical Sciences Instructor’s Guide 4. Chemical Symbols See Table 2 - "List of Elements by For convenience, elements have a symbol which is used as a Name" shorthand for writing the names of elements. The symbol for an element is either one or two letters taken from the name of the element. Note that some have symbols that are based on the historical name of the element. For example, the symbols for silver and gold are Ag and Au respectively. These come from the old Latin names argentum and aurum. The symbol for mercury, Hg, comes from the Greek hydrargyros which means "liquid silver." F. Nomenclature Objective 1.03.06 1. Atomic Number a. The number of protons in the nucleus of an atom. b. All atoms of a particular element have the same atomic number. c. Atomic numbers are integers. d. Atomic number for hydrogen is 1. e. A helium atom has two protons in the nucleus, which means that its atomic number is 2. f. Uranium has 92 protons in the nucleus, and has an atomic number of 92. 2. Mass Number a. The total number of protons plus neutrons in the nucleus of an isotope of an element is called the mass number. b. Since a proton has a mass of 1.0073 amu, we will give it a mass number of 1. c. The mass number for a neutron is also 1, since its mass is 1.0087 amu. d. By adding the number of protons and neutrons we can determine the mass number of the atom of concern. 1.03 - 10 DOE-HDBK-1122-2009 Module 1.03 Physical Sciences Instructor’s Guide (1) A normal hydrogen atom has 1 proton, but no neutrons. Therefore, its mass number is 1. (2) A helium atom has 2 protons and 2 neutrons, which means it has a mass number of 4. (3) If a uranium isotope has 146 neutrons, then it has a mass number of 238 (92 + 146). If it only has 143 neutrons its mass number would be 235. e. The mass number can be used with the name of the element to identify to which isotope of an element we are referring, such as Uranium-235, Uranium-238 (often shortened to U-235 and U-238). 3. Atomic Mass a. The actual mass of a particular isotope. b. The units are expressed in Atomic Mass Units (AMU) (1) AMUs are based on 1/12 of the mass of a carbon-12 atom, which has an atomic mass of 12 amu. (2) The mass of a hydrogen atom is 1.007825 amu (1 proton + 1 electron) (3) The mass of a Uranium-238 atom is 238.0508 and the mass of a U-235 atom is 235.0439. 4. Atomic Weight a. Average weight of an element based on the percent abundance of its naturally occurring isotopes (1) using 13 6 C and 12 6 C (2) 12.00 (0.989) + 13.00 (0.011) = 11.868 + 0.143 = 12.011 amu. b. Units are expressed in AMU c. Used in calculations of chemical reactions 1.03 - 11 DOE-HDBK-1122-2009 Module 1.03 Physical Sciences Instructor’s Guide G. Nuclide Notation 1. Z Ax format where: a. X is the symbol for the element. b. Z is the atomic number - the number of protons. c. A is the mass number - number of protons (Z) plus the number of neutrons (N); therefore, A=Z+N 2. Uranium-238 would be written 238 92U H. Modern Periodic Table 1. The modern Periodic Table is an arrangement of the elements in order of increasing atomic number. A comparison of the properties for selected elements will illustrate that there is a predictable, recurring pattern (periodicity). This observation is summarized in the Periodic Law - the properties of the elements are repetitive or recurring functions of their atomic numbers.

2. Data about each element in the Periodic Table are present in a column and row format. The rows or horizontal sections in the Periodic Table are called periods. The columns or vertical sections in the Periodic Table are called groups or families. 3. The structure of the Periodic Table is directly related to the arrangement of electrons in the atoms. 4. Electrons orbit around the nucleus in structured shells, designated sequentially as 1 through 7 (K through Q) from inside out. Shells represent groups of energy states called orbitals. The higher the energy of the orbital the greater the distance from the nucleus. The lowest energy state is in the innermost shell (K). 5. The number of orbitals in a shell is the square of the shell number (n). The maximum number of electrons which can occupy an orbital is 2. Therefore, each shell can hold a maximum of 2n2 electrons. For example, for the L shell the maximum number of electrons would be 8: L-shell: n = 2 → 2(22) = 8 Objective 1.03.07 See Fig. 5 - "Periodic Table of The Elements" Objective 1.03.08 Objective 1.03.09 See Table 3 - "Electron Configuration of the Elements" See Fig. 4 - "Electron Shells" 1.03 - 12 DOE-HDBK-1122-2009 Module 1.03 Physical Sciences Instructor’s Guide 6. The highest occupied energy level in a ground-state atom is called its valence shell. Therefore, the electrons contained in it are called valence electrons. The rows or periods in the Periodic Table correspond to the electron shells. The elements contained in first period have their valence electrons in the first energy level or K-shell. The elements contained in the second period have their outer or valence shell electrons in the second energy level or L-shell, and so on. 7. The number of electrons in the valence shell determines the chemical properties or "behavior" of the atom. The valence shell can have a maximum of eight electrons, except for the K-shell which can only have two. Atoms are chemically stable when the valence shell has no vacancies; that is, they "prefer" to have a full valence shell. Atoms of elements toward the right of the Periodic Table seem to lack only one or two electrons. These will "look" for ways to gain electrons in order to fill their valence shell. Atoms of elements on the left side of the table seem to have an excess of one or two electrons. These will tend to find ways to lose these excess electrons so that the full lower shell will be the valence shell. 8. The outcome is that certain atoms will combine with other atoms in order to fill their valence shells. This combination that occurs is called a chemical bond, and results in the formation of a molecule. The bond is accomplished by "sharing" or "giving up" valence electrons, thus forming a molecule whose chemical properties are different than those of the individual element atoms. a. Good example - table salt 9. Note the right most column in the Periodic Table. These elements are known as the noble or inert gases because they all have a full valence shell. This means that they "feel" no need to bond with other atoms. Noble gases are thus considered chemically inert and very rarely interact with other elements. 10. The Quantum Mechanical Model Objective 1.03.10 1.03 - 13 DOE-HDBK-1122-2009 Module 1.03 Physical Sciences Instructor’s Guide

a. Over the years, the Bohr model of the atom was found to be inadequate as the principles of quantum mechanics evolved. A newer model, known as the quantum mechanical model, describes the electrons arranged in energy levels corresponding to the "electron shells" of the Bohr model. In the quantum mechanical model the electron is not viewed as particle in a specific orbit, but rather as an electron cloud in which the negative charge of the electron is spread out within the cloud. These energy levels are referred to as orbitals to emphasize that these are not circular "orbits" like those of the Bohr model but rather electron clouds. An electron cloud is a representation of the volume about the nucleus in which an electron of a specific energy is likely to be found. b. The quantum mechanical model further states that the energy levels are subdivided into sublevels, referred to by the letters s, p, d and f. An energy level can contain 1 to 4 sublevels or orbitals, and a maximum of two electrons can reside in each sublevel. For example, the first energy level contains one s sublevel which can accommodate a maximum of two electrons. III. SUMMARY A. Review major topics 1. Physics definitions 2. Law of Conservation of Energy 3. The Atom 4. Periodic Table 5. Valence Electrons B. Review learning objectives IV. EVALUATION Evaluation should consist of a written examination comprised of multiple choice questions. 80% should be the minimum passing criteria for the examination. 1.03 - 14 DOE-HDBK-1122-2009 Module 1.04 Nuclear Physics Instructor’s Guide Course Title: Radiological Control Technician Module Title: Nuclear Physics Module Number: 1.04 Objectives: 1.04.01 Identify the definitions of the following terms: a. Nucleon b. Nuclide c. Isotope 1.04.02 Identify the basic principles of the mass-energy equivalence concept. 1.04.03 Identify the definitions of the following terms: a. Mass defect b. Binding energy c. Binding energy per nucleon 1.04.04 Identify the definitions of the following terms: a. Fission b. Criticality c. Fusion References: 1. "Basic Radiation Protection Technology"; Gollnick, Daniel; 5th ed.; Pacific Radiation Corporation; 2008. 2. "Introduction to Health Physics"; Cember, Herman; 4nd ed.; McGraw-Hill Medical; 2008. 3. ANL-88-26 (1988) "Operational Health Physics Training"; Moe, Harold; Argonne National Laboratory, Chicago. Instructional Aids: 1. Overheads 2. Overhead projector and screen 3. Chalkboard/whiteboard 4. Lessons Learned 1.04 - 1 DOE-HDBK-1122-2009 Module 1.04 Nuclear Physics Instructor’s Guide I. MODULE INTRODUCTION A. Self-Introduction 1. Name 2. Phone number 3. Background 4. Emergency procedure review B. Motivation This lesson is designed to provide an understanding of the forces present within an atom. C. Overview of Lesson 1. Nucleon 2. Nuclide 3. Isotope 4. Mass-Energy Equivalence 5. Mass Defect 6. Binding Energy 7. Fission 8. Criticality 9. Fusion D. Introduce Objectives O.H.: Objectives II. MODULE OUTLINE Objective 1.04.01A. Nuclear Terminology 1. Nucleon - a constituent particle of the nucleus, either a proton or a neutron 2. Nuclide 1.04 - 2 DOE-HDBK-1122-2009 Module 1.04 Nuclear Physics Instructor’s Guide a. Atoms with a specific combination of neutrons and protons b. Nuclides have individual blocks on the Chart of the Nuclides 3. Isotope a. Have the same number of protons but different number of neutrons

b. Same atomic number but different atomic mass number c. Isotopes of Hydrogen have one proton; however, the atomic mass number is different d. Protium (1H) has A=1, deuterium (2H) has A=2, tritium (3H) has A=3 B. Mass - Energy Equivalence Objective 1.04.02 1. Theory on Relativity developed by Albert Einstein in 1905 2. Equation: Write equation on board E = mc2 where: E = Energy m = mass c = speed of light 3. Mass may be transformed to energy and vice versa 4. Mass and energy are interchangeable 5. The mass of an object depends on its speed 6. Matter contains energy by virtue of its mass 7. Energy/Mass cannot be created or destroyed, only converted Information only 8. Pair Annihilation (Mass to Energy example) 1.04 - 3 DOE-HDBK-1122-2009 Module 1.04 Nuclear Physics Instructor’s Guide a. When a positron and electron collide, both particles are annihilated and their mass is converted to energy b. Mass of electron/positron is 0.00054858026 amu, annihilation energy will be: 2(0.00054858026 amu) 931.478 MeV × = 1.022 MeV 1 amu C. Mass Defect/Binding Energy 1. Mass Defect a. Difference between the sum of the protons and neutrons and the actual mass of a nuclide b. Equation: ∗ = (Z)(Mp) + (Z)(Me) + (A-Z)(Mn) - Ma Where: ∗ = mass defect Z = atomic number Mp = mass of a proton (1.00728 amu) Me = mass of a electron (0.000548 amu) A = mass number Mn = mass of a neutron (1.00867 amu) Ma = atomic mass (from Chart of the Nuclides) c. Example for 2 3 Li : 1) A = 7 Z = 3 M = 7.01600 amu 2) Therefore: ∗ = (3)(1.00728) + (3)(0.000548) + (7-3)(1.00867) ­ (7.01600) ∗ = (3.02184) + (0.001644) + (4.03468) - (7.01600) ∗ = (7.058164) - (7.01600) ∗ = 0.042164 amu (1 amu = 931.478 MeV) Objective 1.04.03 See Fig. 1 "Atomic Scale" Work example on board 1.04 - 4 DOE-HDBK-1122-2009 Module 1.04 Nuclear Physics Instructor’s Guide 2. Binding energy a. The energy equivalent of mass defect b. Example for 7 3Li : 0.042164 amu 931.478 MeV BE = × = 39.72 MeV 1 amu 3. Binding energy of a neutron a. Energy added to a nucleus by adding the mass of a single neutron b. Must be calculated for each isotope to determine value c. Example for 235U: Δm = (mn + mU235) - mU236 Δm = (1.00867 + 235.0439) - 236.0456 Δm = 0.0070 amu 0.0070 amu × 931.5 MeV/amu = 6.52 MeV 4. Binding energy per nucleon a. Calculated by dividing the total binding energy of an isotope by its mass number b. Example for 7 3Li : 39.27 MeV = 5.61 MeVpernucleon 7nucleons c. Peaks at about 8.5 MeV for mass numbers 40 – 120 5. Nuclear Transformation Equations (Q Value) Example alpha decay for 226Ra: 226 222 4Ra ⎯⎯ α + Q→ Rn + 88 86 2 D. Terminology 1. Fission Work example on board (1 amu = 931.478 MeV) Work example on board See Fig. 2 "Binding Energy vs. Mass Number" Work example on board Objective 1.04.04 1.04 - 5 DOE-HDBK-1122-2009 Module 1.04 Nuclear Physics Instructor’s Guide a. Splitting of a nucleus into at least two other nuclei with the release of energy b. Two or three neutrons are generally released c. Liquid drop model 1) Equates the nucleus with a drop of water 2) Each contains cohesive forces 3) When forces are overcome, the water drop/atom will split/fission d. Fissile nuclei 1) Neutron binding energy must exceed critical energy for fission 2) Critical energy for fission (Ec): The energy required to drive the nucleus to the point of separation. 3) No kinetic energy required by the neutron

4) Fissile nuclei: 235U, 233U, 239Pu e. Fissionable nuclei 1) Neutron binding energy not enough to exceed critical energy for fission 2) Kinetic energy required to cause fission 3) 238U, 232Th f. Energy released 1) Makes two smaller nuclei from one large nucleus 2) Binding energy per nucleon increases 3) Approximately 200 Mev released per fission (for 235U) g. Fission product See Fig. 3 "Liquid Drop Model of Fission" See Fig. 4 "235U Fission Process" Fertile material is a term used to describe nuclides which generally themselves do not undergo induced fission (fissionable by thermal neutrons) but from which fissile material is generated by neutron absorption and subsequent nuclei conversions. Fertile materials can occur naturally and can be converted into a fissile material by irradiation in a reactor. 1.04 - 6 DOE-HDBK-1122-2009 Module 1.04 Nuclear Physics Instructor’s Guide 1) Created during fission 2) Normally unstable - N/P ratio too high 3) Will undergo radioactive decay until stable - May take less than a second to several hundred years to reach stability 2. Criticality a. Criticality is the condition in which the number of See Fig. 5 "Chain Reaction" neutrons produced by fission is equal to the number of neutrons produced in the previous generation b. The effective multiplication constant or Keff is See Table 1 - "The Effective defined as the ratio of the number of neutrons in the Multiplication Constant" reactor in one generation to the number of neutrons in the previous generation. 1) Subcritical - Keff < 1 2) Critical - Keff = 1 3) Supercritical - Keff > 1 3. Fusion a. Fusion builds atoms b. The process of fusing nuclei into a larger nucleus with an accompanying release of energy c. Change of mass d. Energy released III. SUMMARY A. Review major topics 1. Nucleon 2. Nuclide 3. Isotope 4. Mass-Energy Equivalence 1.04 - 7 DOE-HDBK-1122-2009 Module 1.04 Nuclear Physics Instructor’s Guide 5. Mass Defect 6. Binding Energy 7. Fission 8. Criticality 9. Fusion B. Review learning objectives IV. EVALUATION Evaluation should consist of a written examination comprised of multiple choice questions. 80% should be the minimum passing criteria for the examination. 1.04 - 8 DOE-HDBK-1122-2009 Module 1.05 Sources of Radiation Instructor’s Guide Course Title: Radiological Control Technician Module Title: Sources of Radiation Module Number: 1.05 Objectives: 1.05.01 Identify the following four sources of natural background radiation including the origin, radionuclides, variables, and contribution to exposure. a. Terrestrial b. Cosmic c. Internal Emitters d. Radon 1.05.02 Identify the following four sources of artificially produced radiation and the magnitude of dose received from each. a. Nuclear Fallout b. Medical Exposures c. Consumer Products d. Nuclear Facilities References: 1. "Basic Radiation Protection Technology"; Gollnick, Daniel; 5th ed.; Pacific Radiation Corporation; 2008. 2. ANL-88-26 (1988) "Operational Health Physics Training"; Moe, Harold; Argonne National Laboratory, Chicago. 3. NCRP Report No. 45 "Natural Background Radiation in the United States". 4. NCRP Report No. 56 "Radiation Exposure from Consumer Product Miscellaneous Sources". 5. NCRP Report No. 160 "Ionizing Radiation Exposure of the Population of the United States". Instructional Aids: 1. Overheads 2. Overhead projector/screen 3. Chalkboard/whiteboard 4. Lessons Learned

I. MODULE INTRODUCTION A. Self-Introduction 1. Name 2. Phone number 3. Background 4. Emergency procedure review 1.05 - 1 DOE-HDBK-1122-2009 Module 1.05 Sources of Radiation Instructor’s Guide B. Motivation 1. Radiation sources are not limited to nuclear facilities. The study of sources provides data for: a. Basis for occupational exposures b. Effects from high source exposures c. Assesses impact from nuclear facilities d. Determines use of building materials C. Overview of Lesson 1. Terrestrial radiation 2. Cosmic radiation 3. Internally emitted radiation 4. Radon 5. Nuclear fallout 6. Medical exposures 7. Consumer products 8. Nuclear facilities O.H.: Objectives D. Introduce Objectives 1.05 - 2 DOE-HDBK-1122-2009 Module 1.05 Sources of Radiation Instructor’s Guide II. MODULE OUTLINE A. Natural Background Radiation Sources 1. Terrestrial Radiation a. Earth 1) Source - small amounts of radioactive material found in rock and soil 2) Major isotopes: Uranium and Thorium 3) Exposure dependent on location Atlantic and Gulf coastal = 15-35 mrem/yr Greater U.S. = 35-75 mrem/yr Colorado Plateau = 75-140 mrem/yr 4) Exposure dependent on type of soil Volcanic - 125 mrem/yr Sandstone - 50 mrem/yr Limestone - 25 mrem/yr 5) U.S. average: 1 sq mile 1 ft deep contains 1 ton K-40, 3 tons U-238, 6 tons Th-232 6) Extremely high locations - due to high concentrations of monazite: Kerala India - Population is 70K 16K receive >500 mrem/yr 500 receive >2,000 mrem/yr Highest: 5,865 mrem/yr Minas Garais Brazil Average: 1,160 mrem/yr Max: 12,000 mrem/yr b. Radioactivity in Water 1) All water contains some radioactivity Objective 1.05.01 a. Explain: Different books will use other dose rates. The references used here are NCRP reports. Monazite: a Thorium Mineral 1.05 - 3 DOE-HDBK-1122-2009 Module 1.05 Sources of Radiation Instructor’s Guide 2) Examples Sea water contains K-40 Natural springs contain U and Th Rainwater picks up radioactivity from the air Ground water picks up radioactivity from the soil Contributor to internal doses c. U.S. average of alpha emitters in water is <1 pCi/l 1) Colorado – 40-50 pCi/l 2) Brazil – 240 pCi/l (bottled water) d. U.S. national average from terrestrial (NCRP Report No. 95) is 28 mrem/yr Objective 2. Cosmic Radiation 1.05.01 b. a. Natural radiation originating from outside of our atmosphere b. Discovered during early terrestrial experiments with weather balloons c. Primary 1) Galactic Cosmic Rays From outside the solar system Positively charged particles • 87% protons • 11% alpha • 2% misc. High energies - up to 1020 eV 2) Geomagnetically Trapped When galactic rays approach earth, they must have enough energy to pass through magnetic fields If they lack enough energy, they become trapped in two energy bands 1.05 - 4 DOE-HDBK-1122-2009 Module 1.05 Sources of Radiation Instructor’s Guide • 1K - 3K meters • 12K - 15K meters 3) Solar Cosmic Rays Produced by severe solar flares Consist mainly of protons High energy - detected on ground Low energy - detected at high alt. Measurements • 30,000 ft - 100 mr/hr • 80,000 ft - 10 R/hr Concern for high altitude space travel d. Secondary 1) Results from the interaction of primaries with the earth's atmosphere 2) Cascade effect: one primary ionization = 100 million secondary ionizations 3) Products produced: pions, muons, electrons, photons,

protons, neutrons 4) Primaries absorbed within the upper 10% of the atmosphere 5) Dominant components at ground level are penetrating muons and the electrons they produce. 6) Latitude contributes a small factor due to the earth's magnetic field 7) Exposures increase with altitude, decrease with latitude (Denver 50 mr/yr cosmic) 8) U.S. average = 27 mrem/yr Objective3. Internal Emitters (Food Chain) 1.05.01 c. a. Results from the transfer of natural radiation from the food chain to man 1.05 - 5 DOE-HDBK-1122-2009 Module 1.05 Sources of Radiation Instructor’s Guide b. Deposited internally from trace amounts found in soil, water and air c. Isotopes: Primary - K-40 others - Rb-87, Ra-226, U-238, Po-210, C-14 d. U.S. national average is 39 mrem/yr from Internal Emitters 4. Radon a. Due mostly to Radon and thoron gas b. Radon is a product of the U-238 Series U-238 6 Th-234 6 Pa-234 6 U-234 6 Th-230 6 Ra-226 6 Rn c. Thoron is a product of the Th-232 series Th-232 6 Ra-228 6 Ac-228 6 Th-228 6 Ra-224 6 Rn-220 d. U and Th are present all over. Daughter products diffuse to the surface. These gases attach themselves to dusts and aerosols which are inhaled. e. Radon concentrations are based on amounts of U and Th in the area f. Factors: • Weather (inversions) • Indoor insulation • Ventilation rate g. High Areas: Most has been• Colorado (Grand Junction) – mine tailings removed • Pennsylvania – High radium concentration • Underground mines, caves, caverns, etc. • Helsinki – 240 rem to lung from radon h. U.S. national average for inhaled radionuclides is 200 mrem/yr. 1.05 - 6 DOE-HDBK-1122-2009 Module 1.05 Sources of Radiation Instructor’s Guide Objective B. Man Made Background Radiation Sources 1.05.02 a. 1. Nuclear Fallout a. Refers to the debris that settles as a result of weapons testing b. Contains over 200 fission products, bomb parts and all near-blast matter c. Dispersement is a function of: 1) Bomb yield Kiloton Range - troposphere, 9,000 - 17,000 meters - easily washed down Megaton Range - Stratosphere, may stay aloft for five years 2) Types of blast Surface burst Above ground 3) Meteorological Factors - (weather) d. Weapons test ban treaty of 1962-63 limited testing e. U.S. average from nuclear fallout is <1 mrem/yr (NCRP #93) 2. Medical Exposures Objective a. Diagnostic X-rays 1.05.02 b. 1) Over 300,000 X-ray units in the U.S., about 67% of adult population is exposed each year 2) X-ray machines consist of: a) X-ray tube b) HV Supply c) Filament d) Shielding 3) Three general types: 1.05 - 7 DOE-HDBK-1122-2009 Module 1.05 Sources of Radiation Instructor’s Guide a) Radiography - X-ray tube and a photographic plate (Chest, Dental) b) Fluoroscope - Uses an image intensifier, observes internal processes c) Photofluorographic - Fluorescent screen and camera, large amounts of people b. Medical Radionuclides (Two Types) 1) Nuclear Medicine Used to diagnose medical problems Attaches a radionuclide to a pharmaceutical that will seek a particular organ • Radionuclide selection: • Photon emitter • Short lived Tc-99m, In-113m 2) Radiation Oncology Uses high energy power source to treat tumors Typical 6,000 Curie Co-60 source delivers 100 Rad/min 3) NCRP Report No. 160 gives the dose equivalent for medical radionuclides as ~ 300 mrem/yr 1) computed tomography (total average dose ~ 150 mrem/yr) 2) nuclear medicine (total average dose ~ 75 mrem/yr)

3) radiography/fluroscopy (total average dose ~ 75 mrem/yr) Objective3. Consumer Products (NCRP Report 56) 1.05.02 c. a. Television (Example) 1) Source -- X-rays from High Voltage a few. 2) Limit (1960) -- 0.5 mr/hr at 5 cm 3) 1967 - 149 big screen TVs were recalled, 2 emitted greater than 100 mRad/hr 1.05 - 8 DOE-HDBK-1122-2009 Module 1.05 Sources of Radiation Instructor’s Guide 4) U.S. average -- 0.5 to 1.5 mr/yr b. Shoe fitting fluoroscopes (Example) 1) Source - X-ray tube 2) 1953 - 10,000 in use 3) Exposures - 7 to 14 R per 20 sec exposure GSD - 30 to 170 mr c. Radioluminous Watches (Example) 1) Source - Ra-226 "Glow in the Dark" 2) 10,000,000 still in use 3) Pr-147 and H-3 used today d. Hundreds of other products contribute 1) U.S. national average from all consumer products is 10 mrem/yr 4. Nuclear Facilities a. Public exposures from 1) Mining (several hundred) and milling (20 mills) Objective 2) Fuel Fabrication (21 facilities) 1.05.02 d. 3) Reactors (~90 power, 300 non-power) b. U.S. national average - <1 mrem/yr C. Total Background Radiation The average annual total effective dose to the general population (non­ smokers) from naturally occurring and manmade sources is about 620 mrem. III. SUMMARY A. Review major topics 1. Terrestrial radiation 2. Cosmic radiation 3. Internally emitted 1.05 - 9 DOE-HDBK-1122-2009 Module 1.05 Sources of Radiation Instructor’s Guide 4. Radon 5. Nuclear fallout 6. Medical exposures 7. Consumer products 8. Nuclear facilities B. Review learning objectives IV. EVALUATION Evaluation should consist of a written examination comprised of multiple choice questions. 80% should be the minimum passing criteria for the examination. 1.05 - 10 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide Course Title: Radiological Control Technician Module Title: Radioactivity & Radioactive Decay Module Number: 1.06 Objectives: 1.06.01 Identify how the neutron to proton ratio is related to nuclear stability. 1.06.02 Identify the definition for the following terms: a. radioactivity b. radioactive decay c. 1.06.03 Identify the characteristics of alpha, beta, and gamma radiations. 1.06.04 Given simple equations identify the following radioactive decay modes: a. alpha decay b. beta decay c. positron decay d. electron capture 1.06.05 Identify two aspects associated with the decay of a radioactive nuclide. 1.06.06 Identify differences between natural and artificial radioactivity. 1.06.07 Identify why fission products are unstable. 1.06.08 Identify the three naturally-occurring radioactive families and end product of each. 1.06.09 Given a nuclide, locate its block on the Chart of the Nuclides and identify the following for that nuclide: a. atomic number b. atomic mass c. natural percent abundance d. stability e. half-life f. types and energies of radioactive emissions 1.06.10 Given the Chart of Nuclides, trace the decay of a radioactive nuclide and identify the stable end-product. 1.06 - 1 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide 1.06.11 Identify the definition of the following units: a. curie b. becquerel 1.06.12 Identify the definition of specific activity. 1.06.13 Identify the definition of half-life. 1.06.14 Calculate activity, time of decay, and radiological half life using the formula for radioactive decay.

1.06.15 Identify the definition of the following: a. exposure b. absorbed dose c. equivalent dose d. radiation weighting factor 1.06.16 Identify the definition of the following units: a. roentgen b. rad/gray c. rem/sievert References: 1. "Training Publication 89n, Training Publication 30n"; GPO Division of Radiological Health. 2. ANL-88-26 (1988) "Operational Health Physics Training"; Moe, Harold; Argonne National Laboratory, Chicago. 3. "Health Physics and Radiological Health Handbook"; Shleien; 1992. 4. "Chart of the Nuclides"; Sixteenth Edition, Knolls Atomic; 2003. 5. "Basic Radiation Protection Technology"; Gollnick, Daniel; 5th ed.; Pacific Radiation Corporation; 2008. 6. DOE/HDBK-1019 (January 1993) "Nuclear Physics and Reactor Theory" Volume 1 of 2; DOE Fundamentals Handbook Series. Instructional Aids: 1. Overheads 2. Overhead projector/screen 3. Chalkboard/whiteboard 4. Lessons learned 1.06 - 2 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide I. MODULE INTRODUCTION A. Self Introduction 1. Name 2. Phone number 3. Background 4. Emergency procedure review B. Motivation Atoms consist of protons, neutrons, and electrons and only certain combinations of protons and neutrons exist in nature. Specific combinations of protons and neutrons within the nucleus determine whether nuclei are stable or unstable. Stable nuclei have no excess energy, while unstable nuclei, due to their surplus energy, transform themselves into stable nuclei by giving up energy. The emission of this extra energy to achieve stability is the phenomenon of radioactivity. C. Overview of Lesson 1. Neutron to proton ratio 2. Radioactivity and radioactive decay 3. Radiation characteristics 4. Decay modes 5. Natural/artificial radioactivity 6. Fission product stability 7. Chart of the nuclides 8. Units of activity 9. Activity calculation 10. Measurement terminology D. Introduce Objectives O.H.: Objectives 1.06 - 3 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide II. MODULE OUTLINE A. Nuclear Stability 1. Forces in the Nucleus a. Gravitational Force: very weak attractive force between all nucleons; acts over a relatively long range. b. Electrostatic Force: a strong repulsive force between like charged particles (protons); acts over a relatively long range. c. Nuclear Force: a strong attractive force between all nucleons; acts over an extremely short range. d. In stable atoms the attractive and repulsive forces balance (are equal). If forces are not balanced the nucleus will be unstable. 2. Neutron/Proton Ratio a. Only certain combinations or ratios of neutrons and protons will result in a balance of these forces (stable). b. For atomic numbers less than 20, the neutron to proton ratio (n:p) is about 1:1. c. As number of protons is increased, the electrostatic force increases. The numbers of neutrons must increase more rapidly in order for the nuclear force to balance with electrostatic force. d. As Z increases above 20 the n:p ratio gradually increases until Z = 83, where stable ratio is about 1.5:1. e. There are no completely stable nuclei with Z > 83. f. By graphing the numbers of neutrons against the number of protons for stable isotopes, a "line of stability" is depicted. 3. Stability Ranges a. Nuclear stability is governed by the particular combination of neutrons and protons in a given nucleus.

Objective 1.06.01 See Table 1 - "Forces Acting in the Nucleus" See Fig.1 - "Neutron:Proton Ratios for Stable Nuclides" 1.06 - 4 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay b. Because many elements have several stable isotopes, there is a range for numbers of neutrons that will be stable in a nucleus with a certain number of protons. Instructor’s Guide c. A nuclear arrangement outside of this range will be unstable. An unstable nucleus will attempt to become stable by changing its nuclear configuration. d. Nuclear configuration is changed by eliminating excess neutrons or protons, thereby changing the n:p ratio becoming more stable. B. Radioactivity Objective 1.06.02 1. Definitions a. Nuclear configuration changes occur through transformations. This is done by changing neutrons to protons, or vice versa and then ejecting the surplus mass or energy from the nucleus. b. Particles or energy emitted from the nucleus is called radiation. Radiation can be in the form of particles or waves. c. d. The property of certain radionuclides to spontaneously emit radiation is called radioactivity. In other words, if a nuclide has this property it is said to be radioactive. The term radionuclide has been coined to refer to these "radioactive nuclides." The emission of a particle or electromagnetic radiation in order to reach a more stable configuration produces a change or transformation. The term radionuclide has been coined to refer to these "radioactive nuclides." e. Following a transformation the nucleus is usually more stable than it was, but it may not be completely stable. So, another transformation will take place in which the nucleus will again emit radiation. f. The amount of energy given off and the type of emission that occurs will depend on the configuration of the nucleus immediately before a specific transformation occurs. Each step in the series of transformations will mean a distinct reduction in total mass-energy of the nucleus. 1.06 - 5 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide g. As the energy of the nucleus is reduced, the nucleus is said to disintegrate or decay. The process by which a nucleus spontaneously disintegrates (or is transformed) by one or more discrete energy steps until a stable state is reached is called radioactive decay. h. The nucleus before the decay (or transformation) is called the parent and the nucleus after the decay is called the daughter. i. When there are a series of transformations before a stable state is reached, the daughter of one decay may also be radioactive and thus be the parent to another daughter. j. As the various steps from parent to daughter are traced to stability, a series of transmutations is seen, called a decay chain. The complete chain includes the original parent, all of its daughters and the final, stable endproduct. 2. Nature of Radioactivity a. Certain nuclides are unstable as they occur in nature and are therefore referred to as being naturally radioactive, while others are artificially radioactive because they have become radioactive as a result of some man-made reaction. b. Evidence of natural radioactivity was first reported by Henri Becquerel in 1896. Becquerel demonstrated that uranium ore would darken a photographic plate shielded with opaque paper in much the same manner as X-rays. He postulated that the uranium emitted very penetrating rays, similar to X-rays.

c. The phenomenon ultimately was called radioactivity. In time, it was determined that there were many elements beyond the atomic number of lead (Z=82) which showed similar radiating characteristics. d. After a long and complicated series of investigations, to which many outstanding physicists contributed, a better understanding of natural radioactivity was available. e. The understanding culminated with the experiments of Ernest Rutherford. In 1903, he clearly showed there were three kinds of radioactive emissions, which he named alpha, beta, and gamma, after the first three letters of the Greek alphabet. 1.06 - 6 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide C. Modes of Decay and Types of Radioactive Emissions 1. As mentioned above, Rutherford was initially able to identify three Objective 1.06.03 types of radiation resulting from radioactive decay: alpha, beta and gamma. Initially, all three radiations were commonly referred to as rays. 2. With time, the characteristics of each of these radiations was determined. It was found that alpha and beta are actually particulate radiations, not rays. Since then, other radiations have been discovered through numerous experiments and tests. 3. When a radioactive nuclide decays, a transmutation occurs. The decay product, or daughter has become an atom of a new element with chemical properties entirely unlike the original parent atom. With each transmutation an emission from the nucleus occurs. There are several modes of decay and emissions associated with each mode. 4. Alpha Decay a. With a few exceptions, only relatively heavy radioactive Objective 1.06.04 a. nuclides decay by alpha emission. b. An alpha particle is essentially a helium nucleus. It consists of two protons and two neutrons, giving it a mass of 4 amu. c. Because of the two protons it has an electric charge of +2. d. The symbol α is used to designate alpha particles. e. A nucleus emitting an alpha particle decays to a daughter element, reduced in atomic number (Z) by 2 and reduced in mass number (A) by 4. The standard notation for alpha decay is: f. For example, Radium-226 decays by alpha emission to produce Radon-222 as follows: 1.06 - 7 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide g. Alpha particles are the least penetrating of the three types of radiation. They can be absorbed or stopped by a few centimeters of air or a sheet of paper. 5. Beta Decay Objective 1.06.04 b. a. A nuclide that has an excess number of neutrons (i.e. the n : p ratio is high) will usually decay by beta emission. The intranuclear effect would be the changing of a neutron into a proton, thereby decreasing the n:p ratio, resulting in the emission of a beta particle. b. Beta particles are negatively charged particles. They have the same mass as an electron (1/1836 of proton or 5.49E-4 amu) as well as the same charge (-1) and can be considered high speed electrons. c. Because of the negative charge of the beta particle, beta emission is often more explicitly referred to as "beta-minus" emission (the particle sometimes being referred to as a negatron). d. Beta particles originate in the nucleus, in contrast with ordinary electrons, which exist in orbits around the nucleus. e. The symbol β- is used to designate beta particles. f. In beta-minus emitters, the nucleus of the parent gives off a

negatively charged particle, resulting in a daughter more positive by one unit of charge. Because a neutron has been replaced by a proton, the atomic number increases by one, but the mass number is unchanged. g. In order to conserve energy and momentum between the parent and the daughter plus beta particle there is also the emission of an antineutrino, symbolized by the Greek letter nu with a bar above it ( ) .v h. The standard notation for beta decay is: 1.06 - 8 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay i. For example, Lead-210 decays by beta-minus emission to produce Bismuth-210 as follows: Instructor’s Guide j. Beta particles are emitted with kinetic energies ranging up to the maximum value of the decay energy, Emax. The average energy of beta particles is about 1/3Emax. k. They travel several hundred times the distance of alpha particles in air and require a few millimeters of aluminum to stop them. 1. Neutrinos (ν) and anti-neutrinos ( )v are neutral (uncharged) particles with negligible rest mass, travel at the speed of light and are very non-interacting. They account for the energy distribution among positrons and beta particles from given radionuclides in the positron- and beta-decay processes respectively. 6. Positron Decay Objective 1.06.04 c. a. A nuclide that has a low n : p ratio (too many protons) will tend to decay by positron emission. b. A positron is often mistakenly thought of as a positive electron. If positive electrons existed, then when they encountered an ordinary negative electron, the Coulomb force would cause the two particles to accelerate toward each other. They would collide and then the two equal but opposite charges would mutually cancel. This would leave two neutral electrons. c. Actually, a positron is the anti-particle of an electron. This means that it has the opposite charge (+1) of an electron (or beta particle). Thus, the positron is a positively charged, high-speed particle which originates in the nucleus. d. Because of its positive charge and a rest mass equal to that of a beta particle, a positron is sometimes referred to as "beta-plus." e. The symbol β+ is used to designate positrons. f. With positron emitters, the parent nucleus changes a proton into a neutron and gives off a positively charged particle. 1.06 - 9 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide g. This results in a daughter less positive by one unit of charge. Because a proton has been replaced by a neutron, the atomic number decreases by one and the mass number remains unchanged. h. The emission of a neutrino (symbolized by ν) also occurs in conjunction with the positron emission. i. Positron decay is illustrated by the following notation: j. For example, Nickel-57 decays by positron emission: 7. Electron Capture a. For radionuclides having a low n : p ratio, another mode of decay can occur known as orbital electron capture (EC). b. In this radioactive decay process the nucleus captures an electron from an orbital shell of the atom, usually the K shell, since the electrons in that shell are closest to the nucleus. This mode of decay is frequently referred to as K-capture. The nucleus might conceivably capture an L shell electron, but K electron capture is much more probable. c. The transmutation resembles that of positron emission, as follows: d. The electron combines with a proton to form a neutron, followed by the emission of a neutrino.

e. Electrons from higher energy levels immediately move in to fill the vacancies left in the inner, lower-energy shells. The excess energy emitted in these moves results in a cascade of characteristic X-ray photons. f. Either positron emission or electron capture can be expected in nuclides with a low n : p ratio. g. The intranuclear effect of either mode of decay would be to change a proton into a neutron, thus increasing the n : p ratio. Objective 1.06.04 d. The nucleus might conceivably capture an L shell electron, but K electron capture is much more probable. 1.06 - 10 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide h. Note that 57Ni has two modes of decay. This is an example of branching which is explained in the section DECAY PHENOMENA. 8. Gamma Emission a. Gamma emission is another type of radioactive decay. Nuclear decay reactions resulting in a transmutation generally leave the resultant nucleus in an excited state. Nuclei, thus excited, may reach an unexcited or ground state by emission of a gamma ray. b. Gamma rays are a type of electromagnetic radiation. They behave as small bundles or packets of energy, called photons, and travel at the speed of light. c. The symbol γ is used to designate gamma radiation. d. Since the gamma decay doesn't involve the gain or loss or protons or neutrons, the general equation is slightly different from the other decay equations. For all intents and purposes, gamma radiation is the same as X-rays. Gamma rays are usually of higher energy (MeV), whereas X-rays are usually in the keV range. The basic difference between gamma rays and X- rays is their origin; gamma rays are emitted from the nucleus of unstable atoms, while X-rays originate in the electron shells. The basic difference between gamma rays and visible light is their frequency. All of the transmutation examples given could be accompanied by gamma emission. Although most nuclear decay reactions do have gamma emissions associated with them, there are some radionuclide species which decay by particulate emission with no gamma emission. 1.06 - 11 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide e. Table 2 provides a summary of the characteristics of the various types of radioactive emissions that have been discussed. Table 3 summarizes the various modes of radioactive decay. 9. Other Types of Transformations a. Internal conversion 1) This phenomenon occurs when a gamma photon does not escape the electron cloud surrounding the nucleus, but transfers to one of the orbital electrons enough energy to eject it from the atom. 2) The photon is said to have undergone internal conversion. 3) The conversion electron is ejected from the atom with kinetic energy equal to the gamma energy minus the binding energy of the orbital electron. 4) This process usually takes place in the K-shell. There will then follow emission of characteristic X-rays as with electron capture. b. Isomeric transition 1) Isomeric transition commonly occurs immediately after particle emission; however, the nucleus may remain in an excited state for a measurable period of time before dropping to the ground state at its own characteristic rate. 2) A nucleus that remains in such an excited state is known as an isomer because it is in a metastable state; that is, it differs in energy and behavior from other nuclei with the same atomic number and mass number.

3) The metastable or excited state, is usually represented by a small m following the mass number, A, in the standard nuclide notation. 4) For example, Technetium-99m and Technetium- 99 are isomers. 99mTc will decay 99 Tc with the emission of a 43 43 140.5 keV gamma. See Table 2 - "Types of Radioactive Emissions" and Table 3 - "Modes of Decay" In principle, internal conversion is similar to the photoelectric effect (to be discussed in Lesson 1.07). Generally, the isomer achieves ground state by emitting delayed (usually greater than 10-9 seconds) gamma radiation. 1.06 - 12 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide Further radioactive decay can still occur from the ground state. In this case, 99Tc decays to 99Ru, which is stable. D. Decay Phenomena 1. Each radionuclide, artificial and natural, has its own characteristic pattern of decay. There are several aspects associated with this pattern: a. Modes of decay b. Types of emissions c. Energies of the emissions involved d. Rate of decay 2. Mode of Decay a. All nuclei of a given radionuclide seeking stability by radioactive decay do so in a specific manner. b. There are some radioactive nuclides which may decay with branching, whereby a choice of decay modes exists. In such case, a definite branching ratio exists. A case in point is the decay of 57Ni, mentioned previously. This isotope of nickel decays 50% by K-capture and 50% by β+ emission. The branching ratio would be: 3. Types and energies of Emissions a. Not only do various radionuclides disintegrate in a constant manner insofar as the types of emissions are concerned, but the emissions from each nuclide exhibit a distinct energy picture. Objective 1.06.05 226Ra decays by alpha emission which is accompanied by a gamma photon. This represents the only mode of decay open to 226Ra. 1.06 - 13 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide b. The energies associated with radiations are given in terms of "million electron volts" (MeV). 1) Beta emissions may occur with energies to about 5 MeV 2) Alpha particles to about 10 MeV 3) Gamma photons to about 3 MeV. c. The energy of the particulate radiations is manifested as kinetic energy--the higher the energy the greater the velocity of the particle. d. The velocity of photons is constant (c = speed of light) and energy differences are manifested by varying wavelengths and frequencies. 4. Rate of Decay a. The other characteristic aspect associated with decay patterns is the rate of decay, or activity. The disintegrations of radionuclides occur with a regularity characteristic for each particular species. b. Such disintegrations are spontaneous and random. A single radium nucleus, for instance, may disintegrate at once or wait thousands of years before emitting an alpha particle. c. All that can be predicted with any certainty is that half of all the 226Ra nuclei present will disintegrate in 1,622 years. This period is called the half-life of 226Ra. Half-lives vary greatly for natural occurring radioisotopes; e.g. 212Po, with a half life of 0.298 microseconds and 232Th, with a half-life of over 1.42E10 years. 5. Singly-occurring Natural Radionuclides a. Careful measurements show that almost all materials contain traces of radioactivity.

b. One might suspect that these traces might be due to See Table 4 - contamination with some of the heavy radionuclides belonging "Naturally to one of the radioactive series described. However, some of the occurring lighter elements are themselves weakly radioactive. Radionuclides" 1.06 - 14 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide c. It may have been noted that Carbon-14 was not included as a natural radionuclide in Table 4, even though it has received considerable popular attention in recent years, as naturally- occurring radiocarbon has been found in definite, though small, proportions. The 14C existing in the atmosphere is being formed continually as a result of nuclear reactions between atmospheric nitrogen and neutrons from cosmic rays. This is shown in the following reaction: 6. Artificial Radioactivity Objective 1.06.06 a. There are radionuclides which occur as a result of various man- made reactions. These are called artificial radionuclides. The vast majority of radionuclides are produced in this manner. b. As implied in the nomenclature, natural and artificial radioactivity differ in origin. There are other distinctions between the two types which will be discussed. c. Nevertheless, the nuclei of artificial radionuclides are unstable in much the same manner as their natural counterparts. The intranuclear factors governing decay are also similar for both groups. d. A brief account of the discovery of artificial radioactivity will be given before further discussing its similarities and dissimilarities to natural radioactivity. 7. Induced Transmutations a. In 1919, Lord Rutherford demonstrated that it was possible to produce artificially a transmutation of elements. The manner in which naturally-occurring radioactive atoms are changed or transmuted by emitting radiation has been discussed. b. Lord Rutherford set up and observed a nuclear reaction in reverse, one might say, whereby high-speed charged particles (projectiles) bombarded stable atomic nuclei (target), resulting in a reaction at the nuclear level and inducing a transmutation. 1.06 - 15 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide c. Since the discovery of the transmutation of nitrogen, many hundreds of artificial or induced transmutations have been found. Until 1932, most induced transmutations were performed utilizing naturally occurring alpha emitters as sources of incident particles. d. With the development of particle accelerators, other bombarding particles have been successfully used. No attempt will be made here to catalog the many kinds of possible transmutations, nor will any attempt be made to discuss the theory and quantitative data regarding nuclear reactions. 8. Induced Radioactivity a. During the first 15 years of experimental work with nuclear reactions, the transmutation products (insofar as could be observed) were not radioactive. b. However, the reactions generally were accompanied by the emission of a charged particle and a gamma ray. These emissions are not construed as imparting the property of radioactivity to the target element, since they occur practically instantaneously. c. It was determined in 1934 that induced transmutations could produce nuclei which were residually unstable in somewhat the same manner as naturally occurring radionuclides.

d. Irene Curie and Frederic Joliot reported that certain light elements (boron, magnesium, aluminum), when bombarded with alpha particles, continued to emit radiation for a finite time after bombardment had stopped. 1) The following reaction, involving aluminum bombarded with alpha particles, was the first reported instance of induced or artificial radioactivity: 1.06 - 16 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide 2) The resultant nucleus 30 P was observed to be radioactive, 15 emitting a small charged particle and reaching stability within minutes. e. The work of Curie and Joliot stimulated similar experiments throughout the world. As a result, radioactive isotopes of nearly Over 1,000 unstable every element in the Periodic Table were produced by nuclear species are bombarding a stable isotope with charged particles, neutrons, or listed in the Chart in certain instances photons. of the Nuclides 9. Natural vs. Artificial a. Heavy radionuclides (natural, and artificial) generally decay by a long series of alpha and beta emissions. b. Lighter, artificial radionuclides, such as activation and fission products, usually decay by beta or positron emission or by orbital electron capture. c. In contrast to natural radioactivity, lighter artificially-produced radionuclides generally revert to stability in only a few decay steps. 10. Fission Products a. Another source of radionuclides is nuclear fission. The nuclear fragments directly resulting from fission invariably have too large a proportion of neutrons to protons for stability, and consequently tend to achieve stability by beta minus emission. b. For example, take a thermal fission of 235U: 1) The n:p ratio for stable Cesium (133Cs) is 1.4:1, whereas the above fission product has a ratio of about 1.6:1. 2) The stable ratio for Rubidium (85Rb) is 1.3:1, while the product above has a ratio of about 1.5:1. 3) As can be seen, the fission products in the above equation have too many neutrons. c. Each fission fragment initiates a radioactive series, called a fission decay chain, involving several successive beta decay transformations. 1.06 - 17 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide d. Fission product beta emission, as with other beta emitters, generally is accompanied by gamma emission. i. Predicting Mode of Decay a. Radioactive nuclides tend to decay in a way that results in a daughter nuclide that lies closer to the line of stability. b. Nuclides below the line of stability will usually undergo beta- minus decay. c. Nuclides above the line of stability will usually undergo positron decay or electron capture. d. Nuclides at the upper end of the line of stability will usually undergo alpha decay. e. Figure 2 illustrates the type of decay nuclides in different regions will typically undergo. E. Radioactive Families 1. The transmutations associated with naturally-occurring radionuclides frequently yield a daughter which is also radioactive. 2. To date, about 70 different naturally occurring radionuclides have been identified, each with its own characteristic pattern of radioactivity. 3. Most of these yield radioactive daughters and are now known to be intimately interrelated in radioactive series or families. 4. Three Natural Decay Series a. It has been established that most isolated radioactive species with Z > 82 belong to one of three independent groups or families.

b. Each family starts with a parent radionuclide, decaying or transmuting into a radioactive daughter nuclide, which would again transmute into a daughter nuclide, also radioactive, and so on until stability is attained. These are general rules that have many exceptions, especially in the region of heavy nuclides. Objective 1.06.08 See Fig. 3 - "Natural Decay Series" 1.06 - 18 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide c. One family starts with Uranium-238 ( 238 92 U ) and is called the Uranium series. d. Another starts with Thorium-232 ( 232 90Th ) and is called the Thorium series. e. f. A third starts with Uranium-235 ( 235 92Th ) and is called the Actinium series. In each series, there is a "seesawing" in the transmutation chain between decreasing the atomic number by two with α emission and increasing it by one with β- emission. The historical name of 235U was "Actinouranium." Note its third daughter. g. Each series has an isotope of Radon [historically known as Radon ( 222 86 Rn ), Thoron ( 222 86 Rn ), and Actinon ( 219 86 Rn ) respectively] as a member of the series. All isotopes of Radon are radioactive and are gases at standard temperature and pressure. h. Each series ends in a different stable isotope of Lead ( 206 82 Pb , 208 82 Pb , 207 82 Pb and respectively). i. Figure 3 shows the three natural decay series. 5. Artificial Series a. There is also a fourth series, the Neptunium series, named after its longest-lived member. • Actually, the neptunium series has been artificially produced and no longer occurs in nature, but it is assumed that it did occur in nature at one time and has become extinct because of the relatively short half-lives involved. b. The longest-lived radionuclide in the series is 237 93 Pb with a half- life of 2.2E06 years. c. Assuming the age of the earth is 2.2E09 years, this would indicate that, from the time of creation, 237Np has undergone 1,000 half-lives decay. The fraction of a radionuclide remaining after 1,000 half-lives would be astronomically small—in the order of 10-300 . 1.06 - 19 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide It is obvious, therefore, why it would be difficult to find traces of neptunium and its descendants in nature. F. Chart of the Nuclides a. General Arrangement a. In arranging the nuclides in chart form, the number of neutrons (N) is plotted horizontally on the x-axis against the number of protons (atomic number, Z) on the y-axis b. Such a plot at once reveals the continuity in composition in progressing from the lighter to the heavier elements. Post Chart in classroom or provide books for students to use during discussion. c. The full-size Chart of the Nuclides (poster) is much easier to follow than the Nuclides and Isotopes volume which contains all of the material from the chart in book form. A guide for using the chart is found in the book. b. Specific Nuclide Representation a. Each specific nuclide is represented in the Chart of the Nuclides by a block. b. The coloring and labeling of each block specifies certain information concerning the properties of the nuclide. c. Values for atomic number (Z) are given along the left side of the grid, and values for number of neutrons (N) are found along the bottom. d. A grey block denotes a stable nuclide. A typical example is 11stable sodium ( 23 Na ).

e. Unlike sodium, most elements have more than one stable isotope. For example, magnesium (Mg) has three stable isotopes. Objectives 1.06.09 and 1.06.10 Refer students to block for 23Na. Discuss block color and values, such as percent abundance, activation cross- section and atomic mass. Refer students to blocks for 24Mg, 25Mg and 26Mg. Discuss and compare values. 1.06 - 20 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide f. A white block denotes an artificially produced radioactive nuclide. A typical example is 59 Fe .26 g. A white block with a black triangle in the lower right hand corner denotes an artificially produced radionuclide resulting from slow neutron fission (fission product). h. A grey block with a black bar across the top denotes a long- lived, naturally- occurring radioactive isotope. 238 92 U is a good example. c. Depicting Nuclear Processes a. As a result of decay, radionuclides shift from block to block within the Chart of the Nuclides. The diagram shows the relative locations of the products of various nuclear processes. b. As can be seen, the relative locations (displacements) of the primary modes of decay are: 1) Alpha (α): down 2, left 2 ( , ↓↓ ←← ) 2) Beta (β-): up 1, left 1 ( ↑← ) 3) Positron (β+)/EC: down 1, right 1 ( ↓→ ) c. Displacements can also occur as a result of nuclear reactions brought about through bombarding given nuclides with various nuclear particles or gamma photons. d. Chart of the Nuclides Summary a. The Chart of the Nuclides provides considerable information about the behavior of nuclides. There is continuity in composition of the nuclides. Refer students to block for 59Fe. Discuss half-life, emissions and energies, etc. Refer students to block for 90Sr. Discuss values. Refer students to block for 238U. Discuss block colors and values. Refer students to diagram in Study Guide These changes are depicted in the "Guide for using the Chart of the Nuclides." 1.06 - 21 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide b. A line drawn through the stable nuclides forms a rather smooth curve extending from the lower left to the upper right corner of the Chart of the Nuclides. c. Nuclides below this line are characterized by having an excess of neutrons and will, in general, be beta particle emitters. d. Nuclides above this line are characterized by having an excess of protons and will, in general, decay by positron emission or electron capture. e. Nuclides lying beyond the line of stability will, in general, demonstrate a tendency to seesaw between alpha decay and beta decay. f. All nuclides, if followed through their various decay schemes will eventually end in a gray box (stable isotope). g. The Chart presents in compact style much valuable information concerning the properties of the nuclides. These data include for: 1) Stable nuclides a) Relative abundance b) Cross section for activation 2) Radioactive nuclides a) Types of emissions b) Energies of emissions c) Half-life Objective 1.06.11 G. Units of Activity 1. The rate of decay of a radioactive substance constitutes the quantity of radioactivity, or activity, in that substance. a. The definition of activity refers to the number of transformations (disintegrations) per unit time. b. Since the fundamental unit of time is the second, the quantity activity is measured in disintegrations per second, or dps.

1.06 - 22 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide Since the second is a very short time period in which to make a measurement, activity is measured in units of disintegrations per minutes, or dpm. c. The SI unit of activity is the becquerel, while the historical unit is the curie. Each will be discussed below. 2. The Curie Objective 1.06.11 a. a. Before the large-scale production of artificial radioisotopes, radium had become a standard of comparison for radioactivity measurements. Originally, the unit curie applied only to radium. 1) Named for Marie Curie, it was based on the disintegrations per second (dps) occurring in the quantity of radon gas in equilibrium with one gram of radium. 2) If permitted to attain this equilibrium, one gram of radium will produce about 0.66 mm3 of radon. 3) In this quantity of radon, about 37 billion atoms disintegrate each second. b. In 1930, the International Radium Standard Commission extended the definition to include that quantity of any radioactive decay product of radium which underwent the same number of dps as one gram of radium. • It avoided specifying the figure exactly, so for some years the exact value of the curie varied with each successive refinement in the measurement of the decay constant or the atomic weight of radium. c. In 1950, the International Joint Commission on Standards, Units, and Constants of Radioactivity redefined the curie by accepting 37 billion dps as a curie of radioactivity regardless of its source or characteristics. d. Current regulations define the curie (Ci) as 3.7E10 disintegrations per second (2.22E12 dpm). 1.06 - 23 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide e. Since the curie represents a very large amount of activity, often See Table 5 - smaller, and more convenient subunits are used. "Curie Subunits" 3. The Becquerel Objective 1.06.11 b. a. The SI derived unit of activity is the becquerel (Bq) and is that 1 dps = 1 Bq quantity of radioactive material in which one atom is transformed per second or undergoes one disintegration per second (1 dps). b. Since the becquerel is a rather small unit, metric prefixes are See Table 6 - often applied to aid in designating larger amounts of activity. "Becquerel Superunits" c. The relationship between the becquerel and curie is: 1) 1Bq = 1 dps = 2.7E-11 Ci 2) 1 Ci = 3.7E10 dps = 3.7E10 Bq 4. Using unit analysis and conversion, activity measurements given in dps, dpm or curies can be converted to becquerels. H. Specific Activity Objective 1.06.12 1. Specific activity is defined as the activity per unit mass of a radioactive substance. 2. Reported in units such as curies per gram (Ci/g) or becquerels per kilogram (Bq/kg). 3. Recall that the curie originated from the number of emanations from one gram of radium every second. 4. Thus, the activity of one gram of radium is equivalent to one curie. This means that the specific activity of radium would be 1 Ci/g. 5. It is important, however, to note that when applied to radionuclides other than radium, the unit curie does not make apparent what mass of the material is required. Since one curie of activity is 37 billion dps, the mass of the material required to produce this number of dps will be a function of the decay rate of the atoms of the material (i.e., the disintegration

constant) and of the number of atoms of the material per gram (i.e., gram atomic mass [weight]). 1.06 - 24 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide For example, a curie of pure 60Co (T½ = 5.27 years) would have a mass less than 0.9 milligrams, whereas a curie of natural 238U (T½ = 4.5E9 years) would require over two metric tons of the metal. 6. Obviously, the shorter the half-life of a radionuclide, the greater its specific activity. I. The Radioactive Decay Law 1. The activity of any sample of radioactive material decreases or decays at a fixed rate which is a characteristic of that particular radionuclide. 2. No known physical or chemical agents (such as temperature, pressure, dissolution, or combination) may be made to influence this rate. 3. The rate may be characterized by observing the fraction of activity that remains after successive time intervals. 4. For convenience we choose a fraction that is easy to work with, one- half (½). 5. In using this fraction we can observe the decay of a radionuclide with the passing of time. We can observe how long it takes for the activity to be reduced to one half of the activity. 6. This time that is required for the activity present to be reduced to Objective 1.06.13 one-half we call the half-life. 7. If successive half-lives are observed, we can see a reduction each time by a fraction of one-half, and the effect will be cumulative. a. One half-life reduces to (½)1 b. Two half-lives reduces to ½ × ½ = (½)2 or ¼ c. Three half-lives will reduce to ½ × ½ × ½ = (½)3 or c 8. In the general case the fraction of activity remaining after any number of half lives will be (½)n, where n is the number of half-lives that have elapsed. 9. To put it still another way, the reduction in activity occurs at an exponential rate, which we have expressed as the power of ½. 1.06 - 25 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide 10. In Figure 4 it can be seen that as time passes, radioactive decay occurs at an exponential rate. a. In using the half-life for our time value, we express this exponential function as (½)n. b. Beginning at the instant chosen as the starting point we have 100% of the activity, since no time has elapsed, and the number of half-lives is zero (n = 0). c. If we use t to represent time, at this point, then, t = 0. 11. If we let T½ represent the half-life, then, after one half-life, t = T½, and n = 1. This demonstrates that n represents the ratio of time versus the half-life. 12. Mathematically, this is expressed as: 13. Obviously, the units of t must be the same as the time units of T½ in order to determine the value of n. a. For example, if the half-life of a certain radionuclide is 10 hours, and we allow 4 hours to elapse, the number of half-lives would be 4/10 = 0.4, or 0.4 half-lives. b. The fraction remaining at that instant where t = 4 hours would be: 14. The activity at the instant where t = 0 is the initial or original activity, represented as A0. 15. The activity at any time t after 0 we will denote as At. See Fig. 4 - "Radioactive Decay (Linear Scale)" 1.06 - 26 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide 16. The value of At at any time t will be the fraction remaining times A0. The fraction remaining is determined from the number of half-lives that have passed.

17. Using a proportion we can see the relationship between the two activities: 18. By cross-multiplying we obtain the equation for determining the remaining activity: At = A0(½)n For example, if the initial activity of the radionuclide mentioned above was 52 µCi, then the activity after 4 hours would be: 19. Remember that we stated earlier that radioactive decay is an exponential process. Recall also that a logarithm is, by definition, an exponent. 20. If we were to plot the activity on a logarithmic scale against the time on a linear scale, the resulting curve should be a straight line. 21. This graph shows us that the rate of decay does in fact occur at a constant rate. 22. As time elapses from the starting instant, the activity is reduced thereafter at the constant rate of disintegration for the particular radionuclide involved, which we represent by the Greek letter λ (pronounced "lambda"). 23. In FIGURE 5 the reduction of activity is now a logarithmic (exponential) function of (½)n . A useful "rule of thumb" to remember is that seven half-lives will reduce any activity to less than 1 percent of its original value. See Fig. 5 - "Radioactive Decay (Semi-log Scale)" 1.06 - 27 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide 24. Since n is the ratio of t versus T½, the fraction remaining after time t will be less than 1, resulting in a negative natural-logarithmic value (ln ½ = -ln 2 = -0.693). 25. The fraction remaining will be a function of the decay constant (λ) and the time (t). 26. If we then relate the decay constant to the half-life, λ will be a composite of the natural log of 2 and the half-life. 27. Since the process leads to a decrease in activity, the exponent will be represented by -λt. 28. Therefore, the decay constant itself will represent: 29. Thus, the decay constant is the fraction that disintegrates per unit time (reciprocal time). • If, for example, the half-life is in seconds, λ will be in sec-1. 30. The equation for activity using the decay constant will be: a. Note that in this equation the base of the natural log is raised to a power which includes the -ln 2. b. The result of this equation is exactly the same at that which results from the equation using (½)n . c. It is simply a different way of expressing the decrease in activity with the passage of time as a result of radioactive decay. 31. Using the data in the prior example, the equation would be: Using calculus, the natural logarithm (ln) resulted from the integration of the first equation devised by Rutherford. Objective 1.06.14 1.06 - 28 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide 32. Example: Given 10 mCi of 32P, which has a half-life of 14.2 days, Work example find the quantity remaining after 60 days. problem with students. 33. By algebraic manipulation other variables in this equation can be solved for if the other values are known. Have students work practice problems in • One example would be calculating the original activity based on Study Guide. the current activity, decay constant and elapsed time. J. Series Decay 1. Concerns the mathematical relationship of quantities of activity present when two or more radionuclides exist in a decay chain. 2. Examples of a decay chain are the natural decay series, or a two-step fission product decay series such as: Sr90 → β- + Y90 → β-+ Zr90 (stable)

3. The relationship between three or more radionuclides is described by Bateman. The solution, while straight forward, is quite involved. A two-step relationship (parent-daughter) can be readily derived and is reasonably easy to work with. K. Parent-daughter Relationships 1. In a radioactive decay series, the decay of the parent nuclide produces a daughter product and radiation is emitted. 2. The daughter nuclide also produces radioactivity when it decays, as does each successive daughter in the chain until stability is reached, resulting in total collective activity. 1.06 - 29 DOE-HDBK-1122-2009 Module 1.06 Radioactivity and Radioactive Decay Instructor’s Guide 3. The activity contributed from the parent versus the daughters will vary depending on the half-life of the parent and the half-lives of the daughters. 4. When