NUKEFACT #65

last update July 4, 2001

A similar review for error has been made of DOE-HDBK-1019/1-93, DOE Fundamentals Handbook - Nuclear Physics and Reactor Theory, Volumes 1 and 2. The material in this handbook is used to train reactor operators on DOE nuclear plants, and probably on university reactors which are under the jurisdiction of the DOE. Many of the same serious errors involving basic concepts were found in the DOE handbook. However, it is to the credit of the DOE that they at least provide a standardized text for all facilities, which is more than can be said for the NRC and INPO. This NUKEFACT identifies the technical errors in the DOE Handbook. If you possess a copy of this Handbook, you can use the following listing to flag the errors.

First, five key concepts are discussed which relate to several of the more serious errors. These will be referenced to save repetition when addressing the specific errors. Specific errors are identified by page number and paragraph.

KEY CONCEPTS

1. The Neutron Lifecycle Model: As discussed in NUKEFACT #47 - The Neutron Lifecycle Model the lifecycle model is used to introduce the concept of k_effective, both in terms of the six-factor formula and of the ratio of neutrons in successive generations as shown below:

65.1

Despite the lifecycle model's value as an instructional tool, it is a primitive representation of the reactor because of several inherent assumptions, that are usually not clearly identified. These assumptions include:

1. average (fictional) fission neutrons - no prompt/delayed neutron distinction
   
2. no non-fission neutrons
   
3. duration of life cycle = generation time (weighted avg of prompt/delayed neutron lifetimes)
   
4. constant k-effective
   
5. fission neutrons traverse life cycle as a group - interactions occur in a sequence
   
6. none of generation-1 neutrons survive into generation-2

Another significant retained misconception that arises from Equation 65.1 is that at k_eff = 1.0000 chain reactions are self-sustaining. Again, this is true for the crude life cycle model but is not true in actuality. Since delayed neutrons act as source neutrons to initiate chain reactions, they do not participate in the propagation of each chain. Prompt neutrons propagate chain reactions and, operationally, the reactor is always subcritical on prompt neutrons. As a result, chain reactions are always finite in length. The terminating chains are replaced by new chains initiated by ongoing delayed neutron emissions (see NUKEFACT #4 - Chain Reactions Are NOT Self-Sustaining at Criticality).

Since a generation of fictitious average neutrons is made up of all the prompt neutrons and all the delayed neutrons traversing a particular life cycle, the lifetime of the fictitious average neutrons, called the generation time, must be determined by weighting the lifetimes of the prompt and delayed neutrons by the fraction of each that is in the generation, i.e. in the existing population. A frequently used erroneous definition of generation time is based on the incorrect belief that beta always represents the fraction of delayed neutrons in the total neutron population. Generation time is then determined by weighting the lifetimes of the prompt and delayed neutrons as follows:

	65.2
AN INCORRECT DEFINITION OF NEUTRON GENERATION TIME

where:
l_g = neutron generation time, seconds
(1 - beta) = prompt neutron yield fraction
beta = precursor yield fraction
l_p = the prompt neutron lifetime
l_d = the delayed neutron lifetime

Equation 65.2 indicates that, for a given time in core life, the generation time is a fixed value, a constant. This is not correct. Beta does represent the delayed neutron population fraction for conditions where the neutron level is constant with time, namely for criticality and for equilibrium subcritical multiplication. For any transient power situation, the delayed neutron population differs from beta based on the value of reactivity. For increasing power, the delayed neutron population is less than beta. And for decreasing power, the delayed neutron population is greater than beta.

The correct general expression for the delayed neutron population fraction is (beta - rho), as discussed in NUKEFACT #1 - Beta is NOT The Delayed Neutron Population Fraction. Using (beta - rho) for weighting, the correct expression for neutron generation time is then:

	65.3
CORRECT DEFINTION OF NEUTRON GENERATION TIME

where:
[1 - (beta - rho)] = prompt neutron population fraction
(beta - rho) = delayed neutron population fraction

In common practice, the definition of beta has been corrupted. Beta is not the fraction of the population that consists of delayed neutrons. Beta is the precursor yield fraction.

Also, the term "generation time" applies only to those "fictitious" average neutrons of the lifecycle model. The term generation time was never intended to apply to prompt neutrons alone and it was never intended to apply to delayed neutrons alone. There is no such thing as a "generation time" for prompt neutrons. Likewise, there is no such thing as a "generation time" for delayed neutrons.

2. Source Neutrons: Delayed neutrons - like prompt neutrons, tend to be categorized by the process that creates them. Thus, both prompt and delayed neutrons are considered to be fission neutrons, i.e. both arise from the fission event. This is true. But, what is important is that the delayed neutrons do not serve the same function in the core as the prompt neutrons. The delayed neutrons, like non-fission neutrons, act as a neutron source and initiate chain reactions. Since both the non-fission and delayed neutrons are created through radioactive decay, this is really not too surprising. The prompt neutrons propagate chain reactions. This distinction is paramount in understanding that all reactor behavior, from shutdown to full power, is based on the source multiplication process. Whereas the non-fission source is a constant strength source that becomes insignificant as the reactor is brought critical, the delayed neutron source is a variable strength source that is responsible for the greater part of power increase from shutdown to full power. This important feature of delayed neutrons is lacking in nearly all training resources (see NUKEFACT #2 - Non-Fission Neutrons are NOT the Principal Neutron Source).

3. Reactivity and Reactivity Change: Reactivity is a physical property of the core ... a measure of the nuclear state of one particular reactor condition, in terms of its deviation from criticality and in terms of its ability to reproduce neutrons. Constituents of a reactor, such as control rods, moderator, and structure, do not possess reactivity ... and therefore cannot add reactivity. An existing core reactivity condition may be altered, or changed, but not by adding "reactivity." In addition, adding "negative" anything is usually considered to be removal or subtraction. As an analogy, an existing core temperature may be altered, or changed, but not by adding "temperature."

The difference between two nuclear states is:

65.4

and on rearranging terms:

65.5

Notice that in moving to a new reactivity condition, rho₂, reactivity change, delta-rho, is added to the initial reactivity, rho₁. Or, conversely, notice that reactivity, rho, is definitely not added to the initial reactivity. It is correct to say that a reactivity change has been introduced.

4. Reactivity Coefficients: Two errors are found to be common with respect to reactivity coefficients, including the moderator coefficient, the Doppler coefficient, the void coefficient, and differential rod worth. All coefficients represent "a change in reactivity per unit increase in a parameter", or delta-rho/delta-x where "x" may be 1-degree Fahrenheit, 1-percent void fraction, or 1-inch of rod withdrawal. Coefficients DO NOT represent "reactivity added per unit change in a parameter. The increasing parameter causes a change in the reactivity of the core, where the reactivity change may be either positive or negative. It is the stipulation of an increase in the parameter that sets the algebraic sign of the coefficient.

5. Reactor Period: The equation for stable reactor rate is often expressed incorrectly as:

65.6

where:
T = the reactor period, seconds^-1
rho = reactivity
beta = the precursor yield fraction
lambda = the single precursor group effective decay constant, seconds^-1
l* = l/k
l = the prompt neutron lifetime, seconds
k = the effective multiplication factor

The correct expression as obtained from the in-hour equation is:

65.7

On comparison of Equation 65.7 with Equation 65.6, it is found that Equation 65.7 contains (l*/T) in the numerator of the first term on the right-hand-side. It is this missing term that makes Equation 65.6 incorrect for reactivities approaching prompt criticality and beyond. Equation 65.7 is the correct relationship for reactor period and reactivity (see NUKEFACT #29 - The 1300% Error in Reactor Rate).

SPECIFIC ERRORS

Module 1 - Atomic and Nuclear Physics

p 37: the Figure 11 graph of Activity using a linear scale, does not agree with calculated Activity on page 36.

p 39: the paragraph immediately following Figure 13 contains poor wording, namely "it will increase at a continually decreasing rate"; better said as "it will increase at a decelerating rate."

Module 2 - Reactor Theory (Nuclear Parameters)

p ix: Enabling Objective 3.4 uses incorrect terminology. Neither prompt or delayed neutrons possess generation times. See KEY CONCEPT #1.

p ix: Enabling Objective 3.5 - same error as Enabling Objective 3.4.

p 8: First paragraph after Equation (2-2) - the description of the macroscopic cross section as the effective target area that is presented by all nuclei in one cm³ is incorrect. A later description on page 10 is correct. The macroscopic cross section represents the probability of a neutron undergoing a particular reaction in one centimeter of travel.

p 18: The definition of terms for Equation (2-6) - units on reaction rate (R) are given incorrectly as reactions/sec;(R) represents reactions/sec/cm³

p 29: Enabling Objectives 3.4 and 3.5 are in error. Average fission neutrons in the lifecycle model have a property called the "generation time". Prompt neutrons have a neutron lifetime of the order of 1x10^-4 seconds in commercial reactors. Delayed neutrons have a mean lifetime which is variable, depending on the mix of the precursor inventory. See KEY CONCEPT #1.

p 30: In the paragraph after Table 3, "ell-star" (l^*) is not the generation time for prompt neutrons. l^*, as later used in the equation for reactor period, is the prompt neutron lifetime divided by k_eff. And as indicated previously, the prompt neutrons do not have "generation time" as a physical property. See KEY CONCEPT #1.

p 31: First paragraph - the six delayed neutron groups do not have an "average generation time"; the six delayed groups are generally considered to have a mean life t_m = 1/lambda_eff, which is equal to 12.5 seconds when the neutron level is at steady-state. The mean life of the delayed neutrons is usually taken as the delayed neutron lifetime. See KEY CONCEPT #1

p 31: Second paragraph - there is no such thing as an "average generation time" and the equation given for determining "generation time" is incorrect. See equation in KEY CONCEPT #1.

p 31: In the Example following Equation (2-12), the use of "prompt neutron generation time" and "delayed neutron generation time" is incorrect. See KEY CONCEPT #1.

p 31: In the paragraph before the Summary, change "generation life time" to "generation time." Also note that the 0.0813 second generation time is valid only at the steady-state condition. Also, it is the mean lifetime of the precursors that provides for reactor controllability, not the generation time of fictitious neutrons. See KEY CONCEPT #1.

p 32: Fifth bullet - there is no such thing as a "delayed neutron generation time." See KEY CONCEPT #1.

p 32: Sixth bullet - there is no such thing as a "prompt neutron generation time". See KEY CONCEPT #1.

p 32: Seventh bullet - the equation to calculate the neutron generation time is incorrect. See equation in KEY CONCEPT #1.

Module 3 - Reactor Theory (Nuclear Parameters)

p 4: In the paragraph before the equation for thermal utilization, change "absorbed in any reactor material" to "absorbed in all reactor materials" to be consistent with the equation.

p 5: In the second paragraph, change "core construction material" to "core structural material" for improved clarity.

p 8: The first paragraph under Effective Multiplication Factor, the multiplication factor, whether it be infinite or effective, cannot fully represent reactor behavior because it is based on a primitive model. See KEY CONCEPT #1.

p 8: In the paragraph following the equation for k_eff, a neutron chain reaction is only self sustaining at criticality in the primitive life cycle model where all fission neutrons are fictitious average neutrons. When delayed neutrons are treated separately, they act as source neutrons to initiate chain reactions and the chains are propagated by prompt neutrons. In this real situation a chain reaction is not self sustaining at criticality. All chains are of finite length. Hence, the definition of "prompt criticality" where the chain reaction is self sustaining on prompt neutrons alone. See KEY CONCEPT #1.

p 8: The final paragraph states that when the primitive life cycle model is supercritical the neutron flux increases every generation. This does not have to be the case for an actual reactor, where power can decrease while the reactor is supercritical. Likewise, power can increase in a subcritical reactor. See KEY CONCEPT #1.

p 12: Change the denominator in the equation for thermal utilization from "any" reactor material to "all" reactor material to be consistent with other presentations of this equation, e.g. p 4 and p 15.

p 15: Third, fourth, and fifth bullets - are true for the primitive life cycle model but not for an actual reactor. Chain reactions are not self-sustaining at criticality. Criticality, supercriticality, and subcriticality identify k_eff as being equal to one, greater than one, or less than one, respectively, but in an actual reactor do not necessarily define the neutron population behavior with time. See KEY CONCEPT #1.

p 19: Units of Reactivity - reactivity is a dimensionless number because k_infinity and k_eff are dimensionless numbers, which was not mentioned on p 2 or page 8 of Module 3.

p 21: second paragraph - Reactivity coefficients are not the amount that reactivity will change for a given change in the parameter. Reactivity coefficients are the amount that reactivity will change for a unit "increase" in the magnitude of the parameter, e.g. per degree increase in moderator temperature. This is important because the algebraic sign of the coefficient is determined by the stipulation of "increase" in the parameter. See KEY CONCEPT #4.

p 21: second paragraph - reactivity coefficients are expressed as the "change" in reactivity for a unit increase in the parameter, as expressed by the equation following the second paragraph. Therefore, in the second paragraph the the moderator coefficient represents delta-pcm/deg-F, not pcm/deg-F. See KEY CONCEPT #4.

p 21: Example - the moderator temperature coefficient is incorrectly expressed as -8.2 pcm/deg-F. The coefficient is -8.2 delta-pcm/deg-F. See KEY CONCEPT #4.

p 22: Fifth bullet - the reactivity coefficient is expressed per unit "increase" in the parameter, not change. The use of "magnitude of the parameter change" in the last line is an obscure, and ineffective, way of stating this important requirement. Also note that this bullet refers to the reactivity coefficient in terms of "change in reactivity", not reactivity. It is not possible for the reactivity defect to represent delta-rho, as in the Example on page 21, without the coefficient also representing delta-rho. See KEY CONCEPT #4.

p 26: First, second and third paragraphs - correct the definitions of coefficients to "change in reactivity per degree increase in ... temperature." See KEY CONCEPT #4.

p 27: First and second paragraphs - same comment as p 26.

p 28: First bullet - same comment as p 26.

p 28: Fifth bullet - reactivity is not added; a reactivity change is introduced. See KEY CONCEPT #3. p 30: First paragraph - for correctness and clarity the first sentence should read, "During operation of a reactor at power, the ongoing fissions are sufficient in number to cause the amount of fuel contained in the core to diminish during the time of power production."

p 30: First paragraph - Only the core has the physical property of "reactivity." The magnitude of the core's reactivity may be altered by various means, but none of these means, e.g. control rods, have inherent reactivity of their own. Therefore, they cannot "add reactivity to the core." The neutron absorbing material referred to has no negative reactivity that can be used to balance the excess fuel. A neutron absorbing material can only alter, or change, the core reactivity. Reactivity change may be either positive or negative. See KEY CONCEPT #3.

p 31: First paragraph - same comment as p 30, the burnable poison has no inherent negative reactivity. It can introduce a negative reactivity change into the reactor.

p 31: Second paragraph - same comment as p 30, a negative change in reactivity is introduced by the soluble poison.

p 31: Fourth paragraph - negative is negative. There is no such thing as "less negative." The magnitude of the negative coefficient may be smaller.

p 32: Second paragraph - same comment as p 30, the poison material has no reactivity.

p 36: Equation for equilibrium xenon at the bottom of the page - equilibrium xenon can only exist if the iodine concentration is at equilibrium. To be consistent with the Equation for equilibrium iodine, the equation for equilibrium xenon should contain N_I(eq) and not N_I.

p 37: Last paragraph - xenon introduces a "negative reactivity change", not negative reactivity.

p 38: Figure 5 - The units on the ordinate should be reactivity change, delta-rho, not reactivity (rho or delta-k/k). The numeric values shown on the ordinate are not correct, or consistent with those for samarium shown on Figure 7, p 46. The title of the figure should not be "Xenon Reactivity After Reactor Shutdown". Perhaps a title similar to Figure 7 is better.

p 39: First paragraph - there is no such thing as "negative xenon reactivity" or "positive control rod reactivity."

p 41: First paragraph - it is not the immediate decrease in xenon burnup that causes an increase in xenon concentration. It is a combination of the immediate decrease in xenon burnup (removal) and the continued high rate of xenon production from iodine decay.

p 41: Second bullet - same comment as p 36 concerning N_I(eq) versus N_I

p 42: Fifth bullet - same comment as p 30. There is no reactivity in the control rod.

p 43: First paragraph - samarium is the second most important fission product poison because of a high thermal neutron absorption cross section and a relatively high yield.

p 45: Last paragraph - it is arguable that samarium poisoning is minor when compared to xenon poisoning. Once samarium buildup occurs, it is always in the core, whereas for extended shutdown xenon decays completely. Perhaps this sentence should just be deleted. Perhaps it is based on the error in xenon reactivity change on the ordinate of Figure 5, p 38.

p 46: Figure 7 - same comment as p 38, Figure 5. The units on the ordinate should be reactivity change, delta-rho, not reactivity (rho or delta-k/k).

p 47: First paragraph - same comment as p 30. Helium-3 cannot add negative reactivity; it can only introduce negative reactivity change.

p 49: Types of Control Rods - same comment as p 30. Shim rods cannot remove reactivity and safety rods cannot add negative reactivity.

p 50: Second and third paragraphs - same comment as p 30. A control rod cannot insert negative reactivity.

p 51: First paragraph - states that the resulting reactivity is plotted versus the rod position. This is not correct. Reactivity change, delta-rho or delta-pcm, is plotted versus rod position. The units on the ordinate should be delta-pcm, not pcm. The slope of the curve should then be expressed as delta(delta-rho/delta-in). By substituting specific rho values, it can be seen that delta(delta-rho) is equivalent to delta-rho, being simply the difference between two reactivity values. See KEY CONCEPT #4.

As with other coefficients, control rod worth is based on a specific direction of position change. For control rods it is withdrawal from a previous reference position. Delta-rho with the control rod fully inserted is zero delta-pcm. Integral rod worth values are always positive, by definition.

Wording in the last sentence is poor. Rod withdrawal is not usually expressed in terms of "degree of withdrawal."

p 52: First paragraph - the axial location of greatest neutron flux is not always near the center of the core.

p 52: Second paragraph - differential control rod worth is stated to be "reactivity change per unit movement of a rod". Reactivity change is correct but it is per unit distance of "withdrawal." Differential rod worths and curves are always presented with the worths as positive values, as in Figure 10. The ordinate of Figure 10 is incorrectly shown as rho/in, instead of delta-rho/in. See KEY CONCEPT #4.

p 53: Example 1 - the task is incorrectly stated to be finding the reactivity inserted. The control rod movement introduces a reactivity change in the core. The solution correctly calculates delta-rho. Reactivity and delta-rho are as different as temperature and temperature change. The units on the ordinates of both graphs in Figure 11 should be delta-pcm. It is good practice to always attach an algebraic sign to delta-rho, e.g. +40 delta-pcm.

p 54: Example 2 - same comment as p 30. Reactivity is not inserted by a control rod.

p 54: Solution for Method 1 and Method 2 - the statement that "this is negative because the rod was inserted" is irrelevant if the convention of rod withdrawal is understood. Taking the initial and final conditions as in Method 3 automatically provides the correct algebraic sign. Also note that differential rod worths used are all positive. All results should be in delta-pcm.

p 55: Example 3 - the column of values for delta-rho is not "reactivity inserted" and units are delta-pcm.

p 55: Solution - the column of values for differential rod worth has units of delta-pcm/inch, not pcm/inch.

p 56: Solution continued - the column of values for integral rod worth has units of delta-rho, not reactivity.

p 56: Figure 12 - the units on the ordinate of the graphs are delta-pcm/inch and delta-pcm, respectively.

p 56: Last paragraph - reactivity is not added.

p 57: Second paragraph - poor wording, "positive reactivity ... positive control". Perhaps "adequate control" is better. Reactivity does not "build-in."

p 58: Fourth bullet - integral rod worth is the total reactivity change introduced by withdrawing the rod from the fully inserted position to partially, or fully, withdrawn position. Use of "degree of withdrawal" is not appropriate.

p 58: Fifth bullet - differential rod worth is the reactivity change introduced per unit of withdrawal distance. See KEY CONCEPT #4.

p 58: Sixth bullet - same comment as p 52. The axial location of greatest neutron flux is not always near the center of the core.