NUKEFACT #30

THE EFFECTIVE DECAY CONSTANT DIAGRAM

last update August 20, 1997

Operational reactor behavior is strongly influenced by certain fission fragments, called precursor atoms, which in the process of radioactive decay following fission, emit a neutron. Because of the time delay of precursor neutron emission after the actual fission event, as compared to the near instantaneous prompt neutron emission, these neutrons are called "delayed" neutrons. Precursor atoms are an important factor in operational reactor behavior for the following reasons:

Precursors act as a variable strength neutron source that allows reactor power to be raised to rated output.
The rate of precursor source strength change governs the rate of reactor power change with time.

Thermal neutron fissioning of U-235 nuclei creates a wide variety of fission fragments. Of the 60 plus fission fragments formed from fission, many exhibit decay that produces a neutron emission. For mathematical representation, these precursors are coalesced via similar decay rates into six precursor groups, where the (constant) properties of each group are then an average of its constituent precursors.

PROPERTIES OF THE SIX PRECURSOR GROUPS

Two properties of precursors are of paramount importance to reactor transient behavior, namely the fractional yield from fission of each precursor group, identified by the Greek letter beta, and the decay constant of each precursor group, identified by the Greek letter lambda. For the thermal fission of U-235 these properties, for each of the six precursor groups, are given in the following table.

PRECURSOR GROUP NUMBER	YIELD FRACTION (beta_i)	DECAY CONSTANT (lambda_i) sec^-1	MEAN LIFE (t_{m i}) sec
1	0.0002	0.0127	78.7
2	0.0014	0.0317	31.5
3	0.0012	0.116	8.6
4	0.0027	0.331	3.0
5	0.0008	1.4	0.71
6	0.0002	3.88	0.26

TABLE 30.1 - PROPERTIES of the SIX U-235 PRECURSOR GROUPS

The subscript "i" provides the means for identifying the particular precursor group. The precursor mean life, t_{m i}, in the fourth column is obtained from the relationship t_{m i} = 1/lambda_i . A six precursor group model is commonly used in full scale simulators and other computer programs to simulate actual reactor behavior.

THE SINGLE EFFECTIVE PRECURSOR GROUP

For the purpose of creating a simple chain reaction model conducive to understanding reactor behavior, it is useful to further reduce the precursors to a single group. However, in order to fit the combined six group behavior, beta and lambda of the single group cannot both be of fixed value. Accepted practice is to use a constant precursor fraction, beta, and an adjustable (effective) decay constant, lambda_eff.

beta₂₃₅ = 0.0065 .............................(constant)

lambda_eff = 0.0125 to 0.43 sec^-1 ... (see Figure 30.1)

The subscript "eff" identifies lambda as the single group decay constant. These single effective group parameters are discussed further in the following paragraphs.

PRECURSOR PRODUCTION

From a set of fissions, the fraction of the total fission neutron yield held in the form of precursor atoms is beta, which for U-235 is 0.0065. This value is the sum of the six group lambda-i values in column two of Table 30.1. The yield fraction differs for other fissionable isotopes.

30.1

THE PRECURSOR YIELD FRACTION

where:
beta = the total precursor yield fraction for U-235
precursor atoms = the number of precursor atoms formed from a given set of fission events
prompt neutrons = the number of prompt neutrons formed from a given set of fission events

The average number of precursor atoms produced per fission is:

30.2

where:
nu_d = the average number of precursor atoms from a fission event
nu = the average number of prompt neutrons plus precursor atoms from a fission event

Of the total 2.5 prompt neutrons plus precursor atoms per fission, only about 0.02 neutrons are carried by precursors formed at the instant of fission. And, the production rate of precursor atoms in the entire core is:

core precursor production rate = beta x core fission rate x nu_d ... (atoms/second)

PRECURSOR LOSS

Following fission, the delayed neutrons are gradually released from precursor atoms by radioactive decay. The precursor decay rate is defined by the radioactive "decay constant", as:

30.3

THE PRECURSOR DECAY CONSTANT

This definition of the precursor decay constant applies to the six delay groups listed in column three of Table 30.1. Each of the six precursor groups exhibits a different average rate of radioactive decay, with group one being slowest and group six most rapid.

The actual value of lambda-effective for a single precursor group model can be calculated by choosing a single decay constant that operates on the total precursor inventory to produce a rate of decay that is equal to the actual (six group) decay rate. Thus, the effective decay constant can be derived from:

30.4

THE LAMBDA-EFFECTIVE CONCEPT

Which, on dividing both sides by C_total leads to a concentration weighting of the decay constants for the six precursor groups, as follows:

30.5

THE SINGLE PRECURSOR GROUP LAMBDA-EFFECTIVE

From Equation 30.5 it is obvious that the value of lambda-effective is simply a function of the mix of the six precursor groups. For a constant off-critical reactivity condition (a stable period), the concentrations needed in Equation 30.5 for each of the six groups can be calculated from the following equation:

30.6

THE TRANSIENT PRECURSOR CONCENTRATION

where:
C-bar_i(t) = C_i(t)/(3.1x10¹⁰ x 2.5)
C_i(t) = the core ith-group precursor concentration at time t, atoms
beta_i = the ith-group precursor yield fraction
P(t) = the reactor power at time t
T = the stable reactor period, seconds
lambda_i = the ith-group precursor decay constant, seconds^-1

Note that Equation 30.5, power, P(t), from Equation 30.6 appears in both the numerator and the denominator, thereby cancelling. This is reasonable since only the relative concentrations are important in determining lambda-effective. For generating the Precursor Decay Constant Diagram, the relationship between reactor period and reactivity is readily available from the stable rate equation.

Applying lambda_eff to determine the loss rate of precursor atoms in the entire core, we have:

core precursor loss rate = lambda_eff x core precursor inventory ... (atoms/second)

Of course, the loss of each precursor atom results in the production of a single delayed neutron. The production rate of delayed neutrons (the precursor source strength) is equal in magnitude to the loss rate of precursor atoms.

THE PRECURSOR DECAY CONSTANT DIAGRAM

The precursor decay constant diagram, Figure 30.1, illustrates the dynamic character of lambda-effective for exponential power change with time, as a function of the fuel status, expressed as reactivity.

FIGURE 30.1 -- THE LAMBDA EFFECTIVE DIAGRAM

The vertical scale is lambda_eff in seconds^-1, extending from 0 to 0.5 sec^-1. The horizontal scale is reactivity, extending from -0.0100 to +0.0100. The Reactor Trainer uses Equation 30.5 to calculate the value of lambda-effective for graphic display on this diagram during transient conditions. However, The Trainer does not use lambda-effective in its neutronics calculation, rather the six precursor groups are used.

The S-shaped reference lambda_eff curve shown on Figure 30.1 reflects the differing mix in precursor inventory with the rate of power change. The smaller the decay constant, the slower the precursor decay. The larger the decay constant, the more rapid the precursor decay. The reference curve, extending from the lower left corner to the upper right corner of the diagram represents the locus of lambda-effective values for exponential power change resulting from a constant off-critical reactivity condition. Observe that:

At criticality: the lambda_eff value is approximately 0.08 sec^-1.
Subcritical: as the magnitude of the negative reactivity increases, the value of lambda_eff decreases toward a limit of 0.0127 sec^-1. The limit arises because when far subcritical all groups decay away to insignificance except for the slowest decaying precursor group, which has a decay constant of 0.0127 sec^-1. The mix of precursors degenerates to a single group.
Supercritical: as the magnitude of the positive reactivity increases, the value of lambda_eff increases to a limit of 0.43 sec^-1. The limit arises because when far supercritical the power is rising so rapidly that the only precursors of significance are those just produced in the previous instant. The mix of precursors is equal to that given by the beta-i values in Table 30.1.

The yellow horizontal line at 0.08 seconds^-1, extending from rho = - 0.0100 to criticality is the lambda-effective curve for all equilibrium subcritical multiplication conditions. If the reactor is subcritical and power decays to low levels where the non-fission source becomes significant then lambda-effective will move toward this line.

Figure 30.2 illustrates an actual track of lambda-effective during transient conditions, i.e. while reactivity is changing with time.

FIGURE 30.2 -- TRANSIENT LAMBDA EFFECTIVE BEHAVIOR

The sequence of events is as follows:

Starting at criticality, a continuous reactivity ramp-out is initiated. The track of the transient lambda-effective falls below the reference curve because power is being forced higher by increasing reactivity, which tends to raise the lagging long life precursors to greater concentrations than during exponential increase. A greater number of slow decaying precursors in the mix decreases the value of lambda-effective. The precursor mix is changing continuously during ramp-out, which accounts for the ongoing change in lambda-effective.
On terminating the reactivity ramp at a reactivity of +0.0045, lambda-effective moves vertically upward to the reference curve for exponential power change. With constant reactivity, the precursor mix continues to realign until attaining exponential power increase with time.
From the supercritical condition of rho = +0.0045, a continuous reactivity ramp-in is initiated.The track of the transient lambda-effective falls above the reference curve because power is being forced lower by decreasing reactivity, which tends to reduce the lagging long life precursors to lower concentrations than during exponential decrease. Fewer slow decaying precursors in the mix increases the value of lambda-effective. The precursor mix is changing continuously during ramp-in, which accounts for the ongoing change in lambda-effective.
On terminating the reactivity ramp at a reactivity of -0.0045, lambda-effective moves vertically downward to the reference curve for exponential power change. With constant reactivity, the precursor mix continues to realign until attaining exponential power decrease with time.
From the subcritical condition of rho = -0.0045, a short ramp-out to rho = -0.0035 is made. The purpose of this reacivity change is merely to shift the track so a subsequent lambda-effective increase will be visible and not overlay the previous decrease.
With reactivity constant at rho = -0.0035, the continuing power decrease reaches a level where the non-fission source becomes significant, and as the power levels to equilibrium subcritical multiplication condition, the lambda-effective track moves vertically to the horizontal yellow reference line.

The Lambda-Effective graphics were generated on THE REACTOR TRAINER.

THE RULE-of-THUMB for LAMBDA EFFECTIVE

In reactor operator training programs, the variability of the decay constant during transients is commonly simplified by a rule-of-thumb. For criticality and operational off-critical transients, the rule-of-thumb values for lambda-effective are:

supercriticality ..... lambda_eff = 0.1 seconds^-1
criticality .............. lambda_eff = 0.08 seconds^-1
subcriticality ........ lambda_eff = 0.05 seconds^-1

For reactor transients that are substantially subcritical, such as following reactor scram, the rule-of-thumb may recommend using .... lambda_eff = 0.0127 seconds^-1

The rule-of-thumb is an approximation of the actual lambda-effective value. On comparing the rule-of-thumb values with the reference curve on The Lambda Effective Diagram, Figure 30.1, it can be seen that for the conditions specified, the values are reasonable. The rule-of-thumb provides the student with a convenient means for performing simple calculations involving reactor rate.

SUMMARY

The precursor atoms are important to reactor transient behavior because the precusors act as a variable neutron source which governs the rate of power change with time. When using a single precursor group model, convention is to employ a constant precursor yield fraction and a variable precursor decay rate, as defined by lambda_eff. The lambda_eff value may be obtained by using the rule-of-thumb, or if more precise results are desired, by reading a value of lambda_eff from the reference curve on The Lambda Effective Diagram, for the appropriate reactivity condition.

The student should be aware that the single group precursor decay constant is, in reality, not a constant, but rather a dynamic property that depends on the mix of precursor atoms resulting from the rate and direction of power change.

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