U and Th Half Lives: Why So Long?

[DrDu]

Why do U and Th (and Pu) have so long half lives in comparison with the lighter elements following Bi (i.e. Po, Ra, Rn, Fr, At ..)?

 

[Bill_K]

(This is an interesting point. In the popular mind, Uranium is "highly radioactive". When in fact it is hardly radioactive at all!)

Because Uranium has a low disintegration energy, and the escaping alpha particle has a Coulomb barrier through which it must tunnel. The systematic variation of half-life with disintegration energy is known as the Geiger-Nuttall rule.

 

[DrDu]

(This is an interesting point. In the popular mind, Uranium is "highly radioactive". When in fact it is hardly radioactive at all!)

Because Uranium has a low disintegration energy, and the escaping alpha particle has a Coulomb barrier through which it must tunnel. The systematic variation of half-life with disintegration energy is known as the Geiger-Nuttall rule.

Yes, certainly, but why is the disintegration energy so low? There must be some explanation in models of the nucleus. However, there does not seem to be a magic number nearby.

 

[Bill_K]

Well, it doesn't take anything like a magic number, it turns out to be the result of a rather small effect. Quoting from the book by Krane, a factor of 2 in Q implies a factor of 10²⁴ in half-life. And indeed he gives a plot of Q for heavy nuclei from Os to Lr, and over this entire range, Q only varies from 4 MeV to 9 Mev .

 

[Bill_K]

Q is the disintegration energy, and while the values plotted are experimental data, he also obtains rough agreement from the semiempirical mass formula. For example the formula predicts Q > 0 for heavy nuclei, and a slight trend of decreasing Q for increasing A, constant Z.

 

[DrDu]

Q is the disintegration energy, and while the values plotted are experimental data, he also obtains rough agreement from the semiempirical mass formula. For example the formula predicts Q > 0 for heavy nuclei, and a slight trend of decreasing Q for increasing A, constant Z.

Yes, but the Weitsaecker formula is certainly not accurate enough to explain the stabilty of U and Th.

 

[bcrowell]

I think you do need to invoke shell effects for a full explanation. Otherwise it's hard to imagine any explanation for the oscillating off-on-off-on pattern of alpha decay as you go up in Z from lead. If shell effects were not present, then you would only have liquid-drop physics, and there would only be two effects: (1) a Q value that smoothly increased with Z, and (2) a Coulomb barrier that changed smoothly with Z. I don't see how two such smooth effects could lead to the observed oscillation.

I could be wrong, but I think what's going on is this. The smooth trend is that due to liquid-drop physics, Q values for alpha decay increase smoothly as you go up in Z. Superimposed on top of this, you have some shell effects that become decisive for a few isotopes near the border between alpha-stability and alpha-instability. Lead is stable because of shell effects (magic number 82), so it doesn't alpha decay (Q<0). Stuff a little above lead, such as polonium, can get closer to magic number 82 by alpha decaying, so you get Q>0.

See also: http://en.wikipedia.org/wiki/Island_of_stability

 

[DrDu]

I think you do need to invoke shell effects for a full explanation. Otherwise it's hard to imagine any explanation for the oscillating off-on-off-on pattern of alpha decay as you go up in Z from lead. If shell effects were not present, then you would only have liquid-drop physics, and there would only be two effects: (1) a Q value that smoothly increased with Z, and (2) a Coulomb barrier that changed smoothly with Z. I don't see how two such smooth effects could lead to the observed oscillation.

I could be wrong, but I think what's going on is this. The smooth trend is that due to liquid-drop physics, Q values for alpha decay increase smoothly as you go up in Z. Superimposed on top of this, you have some shell effects that become decisive for a few isotopes near the border between alpha-stability and alpha-instability. Lead is stable because of shell effects (magic number 82), so it doesn't alpha decay (Q<0). Stuff a little above lead, such as polonium, can get closer to magic number 82 by alpha decaying, so you get Q>0.

See also: http://en.wikipedia.org/wiki/Island_of_stability

Yes, the instabilitly of Polonium etc. is clear. Nevertheless I don't understand why stability rises again for Th and U in a region which is far away from magic numbers.

 

[Bill_K]

Suit yourself, DrDu, but I have already given you the reason - it comes from the semiempirical mass formula. Thanks to the amplifying effect of the Coulomb barrier, it only takes a small difference in Q to make an enormous difference in the half-life. I'm not talking about the pairing term, and I'm not talking about shells, although they have an effect too. It's the slow steady variation caused by the volume and surface terms in the mass formula.

And it is not just, as you seem to think, a trend between different elements. It is not just that heavy elements are longer lived than light ones. It is also an isotope effect. Compare the alpha half-lives for the isotopes of one element, Thorium say,

Th-228 is 1.91 y
Th-229 is 7300 y
Th-230 is 75,400 y

 

[DrDu]

Suit yourself, DrDu, but I have already given you the reason - it comes from the semiempirical mass formula.

I would like to believe this, however I am of the same oppinion as bcrowell that the semiempirical formula can only explain a monotonous trend in the Q values for the most stable isotopes.

 

[bcrowell]

For an example of a theoretical approach for understanding this sort of thing, see this paper by Moller and Nix:

http://www.osti.gov/bridge/purl.cover.jsp?purl=/32502-8FXob6/webviewable/32502.PDF

They're using a method called Strutinsky smearing, in which you take a liquid-drop binding energy and add on a shell correction. The original paper by Strutinsky is Nucl. Phys. A122 (1968) 1, and there's a good explanation of the technique in this more recent paper: http://arxiv.org/abs/1004.0079 .

If you look at fig 5 in the Moller and Nix paper, you see a deep minimum in the potential energy landscape at doubly magic 208Pb. Isotopes of Po lie on the steep slope to the right of this minimum. The steep slope corresponds to a high Q value for alpha decay. If, for simplicity, you ignore the N-dependence, then basically you have a function B(Z) that gives the binding energy as a function of Z. Alpha decay rates are sensitive to the slope of this function (B(Z)-B(Z-2)). In the Strutinsky approach, we calculate B by adding two terms, a macroscopic one (liquid drop), plus a microscopic one (Strutinsky correction) which is basically what's shown in fig. 5 (as a function of both N and Z).

Yes, the instabilitly of Polonium etc. is clear. Nevertheless I don't understand why stability rises again for Th and U in a region which is far away from magic numbers.

If you locate 238U (Z=92, N=146) on the plot, it looks like it's in an area where the microscopic correction is relatively flat. This would produce a relatively low Q, explaining why the alpha-decay half-life is so long. If you imagine a toy model in which B(Z) looks like a sine wave, then you would get alpha decay at only one place in each cycle, where dB/dZ was positive.

Suit yourself, DrDu, but I have already given you the reason - it comes from the semiempirical mass formula. Thanks to the amplifying effect of the Coulomb barrier, it only takes a small difference in Q to make an enormous difference in the half-life. I'm not talking about the pairing term, and I'm not talking about shells, although they have an effect too. It's the slow steady variation caused by the volume and surface terms in the mass formula.

You're speaking as though shell effects and Q values were separate considerations. They're not. Part of what determines the Q values is shell effects. I really don't see how you can explain an oscillatory effect based on a "slow steady variation."

And it is not just, as you seem to think, a trend between different elements. It is not just that heavy elements are longer lived than light ones. It is also an isotope effect. Compare the alpha half-lives for the isotopes of one element, Thorium say,

Th-228 is 1.91 y
Th-229 is 7300 y
Th-230 is 75,400 y

I don't think anyone has suggested that only Z matters. It's just that the instability of heavy elements is driven by the Coulomb repulsion, so Z is an important factor in the macroscopic part of the binding energy. If you look at the Moller-Nix figure, it looks like we can make sense out of the thorium data. As you go from 228Th to 230Th (N=138 to N=140), it looks like the shell correction is flattening out, which would tend to make the Q value lower for 230Th.

 

[Bill_K]

You're speaking as though shell effects and Q values were separate considerations. They're not. Part of what determines the Q values is shell effects. I really don't see how you can explain an oscillatory effect based on a "slow steady variation."

An example of what I'm talking about is the plot of Q_α on page four of this. Naturally the shell effect is quite noticeable - near the shell boundaries (N = 126 and Z = 82). But overall the variation is slow and smooth. Pick any Z value and follow the curve - it's slanted downward as A increases, at a nearly steady slope. For the transuranics there's a general upward tendency, but below that, Q_α is remarkably flat. Yes, there's a barely perceptible odd-even staggering, but I don't really see any "oscillatory effect".

 

[phyzguy]

Why do U and Th (and Pu) have so long half lives in comparison with the lighter elements following Bi (i.e. Po, Ra, Rn, Fr, At ..)?

These are far from being the longest half-lives known, and some isotopes of much lighter elements have much longer half-lives. For example, Calcium-48 has a half-life which has been measured at ~4x10^19 years, about 10^10 times longer than U-238. Some half-lives may be even longer. We don't even know for certain whether any elements are absolutely stable, or whether they just have half-lives too long to measure.

 

[bcrowell]

Yes, there's a barely perceptible odd-even staggering, but I don't really see any "oscillatory effect".

Please see my #7 for an explanation of what I mean by that. In more detail, consider what happens as you move up along the line of beta-stability, dealing only with even-even nuclei for simplicity. Alpha decay is absent for Z=82 and lower, present for Z=84-88, very slow for Z=90-92, and then much faster for Z=94 and higher (although it competes with spontaneous fission). The partial half-life for alpha decay goes infinite-short-long-short.

An example of what I'm talking about is the plot of Q_α on page four of this. Naturally the shell effect is quite noticeable - near the shell boundaries (N = 126 and Z = 82). But overall the variation is slow and smooth. Pick any Z value and follow the curve - it's slanted downward as A increases, at a nearly steady slope. For the transuranics there's a general upward tendency, but below that, Q_α is remarkably flat. Yes, there's a barely perceptible odd-even staggering, but I don't really see any "oscillatory effect".

I'm baffled by your characterization of that graph. It's entirely dominated by large up-down oscillations caused by shell effects. It is not true, as you seem to be claiming, that the trend is monotonic for fixed Z. For example, take another look at the bismuth isotopes, which are highlighted in red. The red curve goes high-low-high-low. The most prominent feature is a sharp minimum at A=209, which corresponds the magic neutron number 126.

 

[Bill_K]

Yes Ben, we do see the effect of the closed shell. But do you see that away from the closed shell, Q is monotonic? E.g. the red curve (Bi) falls monotonically from A = whatever it is, 187 I guess, to A = 209. In general the curves are parallel, and equally spaced, and outside the range A = 210 to A = 220 everything is falling. This is about as smooth as you can get.

And this graph shows the actual Q_α, not "microscopic corrections" to it.

 

[DrDu]

An example of what I'm talking about is the plot of Q_α on page four of this.

Thank you Bill for this reference. I found the graph on page 12 even more interesting, as it shows that the maximal half live for a given Z depends mainly on Q. So it is really an effect of Q.

 

[DrDu]

Dear Bill,

yes Strutinski method seems to explain the variation of mass or of D with Z around the liquid drop value. I dug out at home Ring and Schuck "The nuclear many body problem". Fig. 2.32 in this book shows the variation of M with Z around the mass calculated from the liquid drop model. There is a clear maximum around Z=90. The graphic is taken from Nilsson et al., Phys. Lett. 28B, (1969) 458.
Seems that I have to read a little bit.