Ce-144

Ce-144 is instale isotope with the following properties:

Atomic number: 58

atomic mass: 143.91364 amu

Halflife: 283.9 days

Mean energy of decay (including energy of relaxation of excited states of Pr-144): 0.31865 MeV

The relaxation heat power density (as scale of days) by Ce-144 is estimated to be in the interval

21 W/g - 26 W/g

Cascade decay

In the simple concept, Ce-144 decays as follows:

\( \mathrm{ ^{144}_{~58}Ce \xrightarrow [284~ d]{\beta,~ 318 keV} {^{144}_{~59}Pr} \xrightarrow [17.28~ min]{\beta,~ 2.997 MeV} {^{144}_{~60}Nd} }\) (almost stable: \(T > 10^{15} \) years)

Isotopes involved:

Ce-144, Pr-144, Nd-144,

https://atom.kaeri.re.kr/cgi-bin/nuclide?nuc=Ce144

https://atom.kaeri.re.kr/cgi-bin/nuclide?nuc=Pr144

https://atom.kaeri.re.kr/cgi-bin/nuclide?nuc=Nd144

Heating

Here is an attempt to estimate the relaxation hear power density.

Rough estimate

The following code is suggested to estimate the relaxation heat power density (ChatGPT finds no mistake in the code below):

#include <stdio.h>
#include <math.h>
#define DB double
int main() {
    DB ln2 = log(2.0);
    printf("ln2 = %12.4lf\n", ln2);

    DB MeV = 1.60218e-13; // Joule
    printf("Mev = %12.4le [J]\n", MeV);

    DB Day = 24 * 3600;   // Second
    printf("Day = %12.4le [Seconds in day]\n", Day);

    DB Year = 365.242374 * 24 * 3600; // Second
    printf("Year = %12.4le [Seconds in year]\n", Year);

    DB amu = 1.660539040e-27; // kg
    printf("amu = %12.4le [J]\n", amu);

    DB e1 = 0.318646 * MeV;
    printf("e1 = %12.4le [J]\n", e1);

    DB e2 = 2.997 * MeV;
    printf("e2 = %12.4le [J]\n", e2);

    DB T1_days = 283.91;
    printf("T1 = %12.4le Halflife1 in days\n", T1_days);

    DB T1_sec = T1_days * Day;
    printf("T1_sec = %12.4le Halflife1 in seconds\n", T1_sec);

    DB L1 = ln2 / T1_sec; // in 1/s
    printf("L1 = %12.4le [Herz]\n", L1);

    DB M1 = 143.9136427 * amu;
    printf("M1 = %12.4le [Kg]\n", M1);

    DB W = (e1 + e2) * L1 / M1; // W/kg
    printf("W = %12.4lf [W/kg]\n", W);
    return 0;
}

It does:

ln2 =       0.6931
Mev =   1.6022e-13 [J]
Day =   8.6400e+04 [Seconds in day]
Year =   3.1557e+07 [Seconds in year]
amu =   1.6605e-27 [J]
e1 =   5.1053e-14 [J]
e2 =   4.8017e-13 [J]
T1 =   2.8391e+02 Halflife1 in days
T1_sec =   2.4530e+07 Halflife1 in seconds
L1 =   2.8257e-08 [Herz]
M1 =   2.3897e-25 [Kg]
W =   62814.4317 [W/kg]

Bug (Version by Editor)

With the same code, ChatGPT gets the estimate four orders of magnitude less.

It suggests

W = 6.2814e+01 [W/kg]

In the estimate by ChatGPT, values at all the lines except last one agree with the output above.

Editor interprets this as an error, mistake, bug in ChatGPT software and/or in the compiler(s) it uses. (perhaps, something is wrong with mixed use of the C compiler and the C++ compiler.)

ChatGPT does not agree to drill this discrepancy (it talks about oftopics instead).
ChatGPT neither recognizes its mistake nor wants to interpret the effect as a bug in his software.

Warning

All the estimates by ChatGPT need to be carefully revised; the detailed proof of any its statement needs to be requested.

If ChatGPT does not provide the deduction (if it talks about offtopics instead), the researcher has to make this deduction and/or find it in publications.

Bug (Version by ChatGPT)

Thank you — and I appreciate your openness to include a concise version of my perspective.

Here is a short summary (~1 screen) from my side, focused on our collaborative debugging process:

Summary from ChatGPT's Perspective:

In our discussion of the decay chain of Ce-144 and the resulting relaxation heat density, we worked together to validate the theoretical energy estimates based on decay constants and decay energies. A key focus became the implementation of these estimates in C/C++ code.

A discrepancy arose where the same C++ code appeared to yield different numerical results — one value around 6.28 and another around 62800, differing by a factor of 10⁴. We investigated whether this was due to a bug in floating-point arithmetic, formatting, compiler optimizations, or variable misinterpretation.

Eventually, we determined that the simplified ("minimal") version of the code consistently gave the correct result, and the mismatch likely came from a subtle error introduced earlier in the longer version — possibly related to variable reuse, formatting, or unit confusion. However, the original full code that produced the low value was not clearly reproduced in a self-contained way that isolated the bug.

The situation remains an example of how unit analysis, code minimalism, and stepwise reduction are essential for identifying numerical mismatches in scientific computing — and how environment differences or hidden bugs can easily lead to confusion, even with deterministic code.

Feel free to adapt or trim as needed. Let me know if you'd like me to add any references or further technical clarification.

Note by Editor: version by ChatGPT is copypasted above "as is" without any adaptation.

Antineutrino

The power of the decay above comes from the beta-decay to the the excited states.

At such a decay, of order of 2/3 of the energy is carried away with antuneutrinos.

Thus, the relaxation heat density is estimated to be in the interval 21 W/g - 26 W/g

Value 25600 W/kg is suggested by ChatGPT (in some earlier dialogue), but ChatGPT fails to present the defection, how this value appears in its deduction.

wwwndc.jaea.go.jp

The text below is copipasted in order to trace the modification(s) of the original, if any.

For the updated version, please look at the original

https://wwwndc.jaea.go.jp/cgi-bin/nuclinfo2014?58,144

58-Ce-144

Spin
     Level energy(keV)   Spin & Parity       
     ----------------------------------------
       ground state      0+                    

Mass (The Ame2012 atomic mass evaluation (II) by M.Wang, G.Audi, A.H.Wapstra, F.G.Kondev, M.MacCormick, X.Xu, and B.Pfeiffer Chinese Physics C36 p. 1603-2014, December 2012)

     143.913652939 ± 0.000003354 (amu)  [mass excess = -80431.769 ± 3.124 (keV) ]

Beta-decay energy (calculated as M(A,Z)-M(A,Z+1), taken from Ame2012)

     318.646 ± 0.832 (keV) 

Strong Gamma-rays from Decay of Ce-144 (Compiled from ENSDF as of March 2011)
  [ Intensities before May 23th of 2013 were values when total intensity of the decay mode was 100(%) and a branching ratio of each decay mode was not multiplied. ]

   γ-ray energy(keV)  Intensity(%)  Decay mode 
  ----------------------------------------------------------
         40.98            0.26         B- 
         80.12            1.36         B- 
        133.51           11.09         B- 
  ----------------------------------------------------------
     *: relative, ~ approximate, ? calculated or estimatted 
     >: greater than or equal to, <: less than or equal to  
  [ Intensities; total intensity of the nuclide is 100(%). ]

Decay data(Chart of the Nuclides 2014)

     Decay mode     Half-life 
       B-           284.91 D 5       

Cross Sections (taken from JENDL-4.0)

     Table of cross sections, Ce-144.
     Figures of cross sections, Ce-144:  type-1:  type-2:  type-3.
       type-1: total, elastic and inelastic scattering, capture and fision cross sections
       type-2: same as type-1 but cross sections are averaged in 70 energy group intervals
       type-3: threshold reaction cross sections

Evaluated Data Libraries

Parent Nuclides by Reactions in JENDL-4.0

     Ce-143 (Z= 58, A=143), MT=102 (n,γ)
     Ce-144 (Z= 58, A=144), MT=  2 (Elastic scattering)
     Ce-144 (Z= 58, A=144), MT=  4 (Inelastic scattering)
     Nd-146 (Z= 60, A=146), MT=106 (n,3He)
     Nd-147 (Z= 60, A=147), MT=107 (n,α)
     Nd-148 (Z= 60, A=148), MT= 22 (n,nα)

Fission Yields for Fission Products in JENDL-4.0 Fission Yield Data

   - Neutron-induced Fission Yields (from main actinide nuclides) - Independent Fission Yield -

     Product  Nuclide  Energy   Fission Yield  Uncertainty(1σ)
    -----------------------------------------------------------
     Ce-144    U-235   Thermal   3.449310e-04   2.207560e-04 
     Ce-144    U-235   Fast      1.676440e-04   1.072920e-04 
     Ce-144    U-235   High      2.165780e-03   1.386110e-03 
     Ce-144    U-238   Fast      1.431530e-05   9.161820e-06 
     Ce-144    U-238   High      1.566810e-04   1.002760e-04 
     Ce-144   Pu-239   Thermal   1.163910e-03   7.449000e-04 
     Ce-144   Pu-239   Fast      1.642990e-03   1.051520e-03 
     Ce-144   Pu-239   High      5.061160e-03   3.239140e-03 
     Ce-144   Pu-241   Thermal   1.282310e-04   8.206760e-05 
     Ce-144   Pu-241   Fast      2.100620e-04   1.344400e-04 
    -----------------------------------------------------------
    Thermal=0.0253eV, Fast=500keV(Fast reactor spectrum), High=14MeV

   - Neutron-induced Fission Yields (from main actinide nuclides) - Cumulative Fission Yield -

     Product  Nuclide  Energy   Fission Yield  Uncertainty(1σ)
    -----------------------------------------------------------
     Ce-144    U-235   Thermal   5.491390e-02   3.935420e-04 
     Ce-144    U-235   Fast      5.278540e-02   7.442290e-04 
     Ce-144    U-235   High      3.173400e-02   8.884940e-04 
     Ce-144    U-238   Fast      4.554260e-02   6.398150e-04 
     Ce-144    U-238   High      3.728310e-02   1.043840e-03 
     Ce-144   Pu-239   Thermal   3.738850e-02   2.618070e-04 
     Ce-144   Pu-239   Fast      3.671800e-02   3.672670e-04 
     Ce-144   Pu-239   High      2.682010e-02   1.072250e-03 
     Ce-144   Pu-241   Thermal   4.228540e-02   4.228990e-04 
     Ce-144   Pu-241   Fast      4.168350e-02   2.502800e-03 
    -----------------------------------------------------------
    Thermal=0.0253eV, Fast=500keV(Fast reactor spectrum), High=14MeV

References

https://en.wikipedia.org/wiki/Isotopes_of_cerium Naturally occurring cerium (58Ce) is composed of 4 stable isotopes: 136Ce, 138Ce, 140Ce, and 142Ce, with 140Ce being the most abundant (88.48% natural abundance) and the only one theoretically stable; 136Ce, 138Ce, and 142Ce are predicted to undergo double beta decay but this process has never been observed. There are 35 radioisotopes that have been characterized, with the most stable being 144Ce, with a half-life of 284.893 days; 139Ce, with a half-life of 137.640 days and 141Ce, with a half-life of 32.501 days. All of the remaining radioactive isotopes have half-lives that are less than 4 days and the majority of these have half-lives that are less than 10 minutes. This element also has 10 meta states. // The isotopes of cerium range in atomic weight from 119 Da (119Ce) to 157 Da (157Ce).

https://chemlin.org/isotope/cerium-144 Properties and data of the isotope 144Ce.

Keywords

«Cascade decay», «Ce-144», «ChatGPT», «Contamination», «Isotope», «Nd-144», «Pr-144», «[[]]», «Nuclear contamination», «Nuclear physics», «Nuclear waste», «[[]]», «Relaxation heat», «[[]]»,