In the United States, mid-20^{th} century – present, a unit of reactivity used in nuclear engineering. One inhour is the reactivity of a critical reactor in which neutron flux increases (positive value) or decreases (negative value) by a factor of *e* in 1 hour.

It is sometimes said that the reactivity in inhours is equal to the reciprocal of the reactor period in hours, but this is true only in certain circumstances, see below.

References to the paper by Nordheim cited below say that it contains an inhour formula, and hence presumably a definition of the inhour. If so, it anticipates by more than a year the earliest definition of the unit in a generally circulated journal. Unfortunately, so far we have been unable to locate a copy of Nordheim's paper.

L[othar] W[olfgang] Nordheim.

Manhattan District Declassified Document No. 35, June 14, 1946

Equation 3.

The Argonne researchers on 2 December 1946. The men in the front row are the authors of the paper; from left to right, Fermi, Zinn, Wattenberg and Anderson.

Courtesy Argonne National Laboratory

In 1947 Fermi and his colleagues defined the inhour in reporting an investigation of neutron absorbers. They describe it as a “unit of rod position,” the distances (varying with the rod's location) that the control rod must be moved to produce a certain change in reactivity.

H[erbert] L. Anderson, E[nrico] Fermi, A[lbert] Wattenberg, G[eorge] L. Weil and W[alter] H. Zinn.

Method for Measuring Neutron-Absorption Cross Sections by the Effect on the
Reactivity of a Chain-Reacting Pile.

*Physical Review,* volume **72**, number 1, pages 16 - 24 (July 1, 1947).

From page 17

In this work, we have measured the effect of the insertion of an
absorber in the pile by observing the displacement of the control rod which was
required to return the pile to the critical condition. It was found, however,
that for a given change in reactivity, the displacement required, if measured in
centimeters, was not always the same but depended on the position of the control
rod in the pile. It was convenient to introduce a new unit of rod position such
that displacements measured in the new units would always be proportional to (*k*
⁻¹). In order to provide a precise definition of such a unit, use was made of
the property of the pile that its period is proportional to 1/(*k* − 1) in the limit of long periods. Accordingly, the unit of rod position was given the name *
inhour* (from “inverse hour,” symbol: ih), with the significance that
when the control rod is displaced from the critical position by 1 inhour, the
pile will have a period of (very nearly) 1 hour.

From pages 21 & 22

The inhour is proportional to * k* − 1, thus,

ih = C(*k * − 1)

where the value of C is given by

and T is the period of the pile in hours.

The value of the inhour is measured from its zero value corresponding to control rod out of the pile.

The inhour is useful as a measure of rod displacement because it is a measure of pile reactivity which is independent of the position of the control rod, so that linear interpolations are accurate in comparing the effects of different absorbers. Moreover, its value has an almost unambiguous significance for all graphite-uranium piles.

Six months later another paper, also by workers at the Argonne National Laboratory, describes the inhour as “the pile reactivity unit,” instead of a “unit of rod position,” and makes a small correction:

D. J. Hughes, J. Dabbs, A. Cahn and D. Hall.

Delayed Neutrons from Fission of U-235.

*Physical Review*, volume **73**, number 2, pages 111-125, (January 15,
1948).

From page 111 (abstract)

...the value of the inhour (the pile reactivity unit) is calculated.

From pages 123 & 124

Observations with graphite-uranium piles have shown that the
amount of reactivity (“excess *k*”) which will give a pile period
of one hour is 2.5±0.5×10⁻⁵. This amount of reactivity is by
definition one “inhour.” ...

The periods and yields reported here also make it possible to
give a more accurate formulation of the relationship between the reactivity in
inhours and the pile period (the “inhour formula”) than that given in
reference 10 [*Anderson *et al *above*]. Substitution of the new
values into the formula given in Eq. 3 of reference 11 [*Nordheim, cited above*]
gives the following relationship:

and the two formulas (old and new) are compared in Fig. 14. It
is to be noted that the present formula is adjusted to equal unity for *T*
= 3600 in accordance with the definition of the inhour, while the formula of
reference 10 [*Anderson * et al] (aside from the older yield and period values)
differ slightly in that the sum of the numerators is set to equal 3600 and,
hence, the reactivity is slightly less than one inhour for a period of one hour.

By the 1980's

Samuel Glasstone and Alexander Sesonske.

*Nuclear Reactor Engineering*. 3rd edition.

New York: Van Nostrand Reinhold, 1981.

From page 247

The Inhour Formula

**5.48.** The reactivity is sometimes expressed in terms
of the inverse hour or “inhour” unit, defined as the reactivity which
will make the stable reactor period equal to 1 hour, i.e., 3600 s [*cites
Anderson *et al*, as above*]. Thus the
value of the inhour unit in terms of reactivity is obtained by setting *T _{p}* in
equation (5.37) equal to 3600 s, since the

(5.39)

[*where*

*l*, neutron generation time

*T _{p}*, reactor period

*β _{i}*, fraction of delayed neutrons of the

*λ _{i}*, precursor decay constant (

*Note that the “6” makes this formula specific to
U-235 reactors.*]

This expression is the correct form of the inhour formula, although the related equations (5.16) and (5.37) are sometimes referred to by this name.

[*those equations:*

(5.16)

(5.37) ]

**5.49.** It can be seen from equation (5.39) that, in general, the reactivity given in inhours does not define the stable reactor
period in a simple manner. Nevertheless, for large reactors, such as the
natural uranium-graphite reactors, the reactivity is frequently expressed
in inhours. The reason is that for small reactivities, i.e., for long reactor
periods, unity may be neglected in comparison with λ_{i}T_{p} and 3600λ* _{i}*, and equation (5.39) then reduces to

(5.40)

so that the reactivity in inhours is inversely related to the stable reactor period in a simple manner. In fact, if the reactor period is expressed in hours, it is equal to the reciprocal of the reactivity in inhours.

**5.50.** By comparing equations (5.38) and (5.40), a
relationship between * ρ* and Ih may be obtained, namely,

For uranium-235, the sum of the β_{i}/λ* _{i}* values is 0.084 (table 5.2), and so

ρ ≈ 2.3 × 10⁻⁵ Ih.

Although this result is strictly applicable only for long reactor periods, it is sometimes combined with equation (5.37) to give the relationship

The times *l* and *T _{p}* are usually expressed in seconds, and so the

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