Gamow factor

The probability of two nuclear particles overcoming their electrostatic barriers is given by the following factor:{{cite web|publisher=Dept. Physics & Astronomy University College London|title=Nuclear reactions in stars|url=https://zuserver2.star.ucl.ac.uk/~idh/PHAS2112/Lectures/Current/Part7.pdf|access-date=2014-11-12|archive-date=2017-01-15|archive-url=https://web.archive.org/web/20170115214447/https://zuserver2.star.ucl.ac.uk/~idh/PHAS2112/Lectures/Current/Part7.pdf|url-status=dead}}

: $P\_\backslash text\{G\}(E)\; =\; e^\{-\backslash sqrt\{\{E\_\backslash text\{G\}\}/\{E\}\}\},$

where $E\_\backslash text\{G\}$ is the Gamow energy

: $E\_\backslash text\{G\}\; \backslash equiv\; 2\; \backslash mu\; c^2\; (\backslash pi\; \backslash alpha\; Z\_\backslash text\{a\}\; Z\_\backslash text\{b\})^2,$

where $\backslash mu=\; \backslash frac\{m\_\backslash text\{a\}\; m\_\backslash text\{b\}\}\{m\_\backslash text\{a\}\; +\; m\_\backslash text\{b\}\}$ is the reduced mass of the two particles.{{efn|Identical (protons, ₂He²⁺):$\backslash frac\{1\}\{2\}m,\; E\_G=mc^2(\backslash pi\backslash alpha\backslash cdot(1,4))^2=m(c\backslash pi\backslash alpha)^2\backslash cdot(1,16).$$m\_a\backslash gg\; m\_b:\backslash mu\backslash lesssim\; m\_b,$ cation vs ₁H¹⁺}} The constant $\backslash alpha$ is the fine-structure constant, $c$ is the speed of light, and $Z\_\backslash text\{a\}$ and $Z\_\backslash text\{b\}$ are the respective atomic numbers of each particle.

It is sometimes rewritten using the Sommerfeld parameter {{math|η}}, such that

: $P\_\backslash text\{G\}(E)\; =\; e^\{-2\backslash pi\; \backslash eta\},$

where {{math|η}} is a dimensionless quantity used in nuclear astrophysics in the calculation of reaction rates between two nuclei and it also appears in the definition of the astrophysical S-factor. It is defined as{{cite book |last1=Rolfs |first1=C.E. |url=https://books.google.com/books?id=BHKLFPUS1RcC&pg=PA156 |title=Cauldrons in the Cosmos |last2=Rodney |first2=W.S. |publisher=University of Chicago press |year=1988 |isbn=0-226-72456-5 |location=Chicago |page=156}}{{cite journal |last=Breit |first=G. |year=1967 |title=Virtual Coulomb Excitation in Nucleon Transfer |url=http://www.pnas.org/content/57/4/849.full.pdf |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=57 |issue=4 |pages=849–855 |bibcode=1967PNAS...57..849B |doi=10.1073/pnas.57.4.849 |pmc=224623 |pmid=16591541 |access-date=27 January 2015 |doi-access=free}}

: $\backslash eta\; =\; \backslash frac\{Z\_a\; Z\_b\; e^2\}\{4\; \backslash pi\; \backslash epsilon\_0\; \backslash hbar\; v\}\; =\; \backslash alpha\; Z\_1\; Z\_2\; \backslash sqrt\{\backslash frac\{\backslash mu\; c^2\}\{2E\}\},$

where {{math|e}} is the elementary charge, {{math|v}} is the magnitude of the relative incident velocity in the centre-of-mass frame.{{efn|$\backslash eta\_\{id\}=\backslash frac\{(1,2)^2e^2\}\{4\backslash pi\backslash epsilon\_0\backslash hbar\; v\}=\backslash frac\{1\}\{2\}c\backslash alpha\backslash cdot(1,4)\backslash sqrt\{\backslash frac\{m\}\{E\}\}$}}

File:Scanning tunneling microscope - rectangular potential barrier model.svg

The derivation consists in the one-dimensional case of quantum tunnelling using the WKB approximation.[http://web.ihep.su/dbserv/compas/src/gamow28/eng.pdf Quantum Theory of the Atomic Nucleus, G. Gamow]. Translated to English from: G. Gamow, ZP, 51, 204 Considering a wave function of a particle of mass m, we take area 1 to be where a wave is emitted, area 2 the potential barrier which has height V and width l (at k [m^-1] and energy E we get:

: $\backslash Psi\_1=Ae^\{i(kx+\backslash alpha)\}e^\{-i\{Et\}/\{\backslash hbar\}\}$

: $\backslash Psi\_2=B\_1e^\{-k\text{'}x\}+B\_2e^\{k\text{'}x\}$

: $\backslash Psi\_3=(C\_1e^\{-i(kx+\backslash beta)\}+C\_2e^\{i(kx+\backslash beta\text{'})\})\backslash cdot\; e^\{-i\{Et\}/\{\backslash hbar\}\}$

where $k\; =\; \backslash sqrt\{2mE/\backslash hbar^2\}$ and $k\text{'}\; =\; \backslash sqrt\{2m(V-E)/\backslash hbar^2\},$ both in [1/m]. This is solved for given A and phase α by taking the boundary conditions at the barrier edges, at $x=0$ and $x=l$: there $\backslash Psi\_\{1,3\}(t)$ and its derivatives must be equal on both sides. For $k\text{'}l\; \backslash gg\; 1$, this is easily solved by ignoring the time exponential and considering the real part alone (the imaginary part has the same behaviour). We get, up to factors

• depending on the β phases which are typically of order 1, and
• of the order of $\{k\}/\{k\text{'}\}=\backslash sqrt\{\{E\}/\{(V-E)\}\}$ (assumed not very large, since V is greater than E (not marginally)):

$\backslash Psi\_1=Ae^\{i(kx+\backslash alpha)\}\; ,\; \backslash Psi\_3=C\_1e^\{-i(kx+\backslash beta)\}+C\_2e^\{i(kx+\backslash beta\text{'})\},$

$\backslash Psi\_2\backslash approx\; Ae^\{-k\text{'}x\}\; +Ae^\{k\text{'}x\}:\; B\_1,B\_2\backslash approx\; A$ and $C\_1,\; C\_2\; \backslash approx\; \backslash frac\{1\}\{2\}A\backslash frac\{k\text{'}\}\{k\}\; e^\{k\text{'}l\}.$

Next, the alpha decay can be modelled as a symmetric one-dimensional problem, with a standing wave between two symmetric potential barriers at

File:Potential-barrier-for-alpha-particle (simple).svg

Due to the symmetry of the problem, the emitting waves on both sides must have equal amplitudes (A), but their phases (α) may be different. This gives a single extra parameter; however, gluing the two solutions at $x=0$ requires two boundary conditions (for both the wave function and its derivative), so in general there is no solution. In particular, re-writing $\backslash Psi\_3$ (after translation by $q\_0$) as a sum of a cosine and a sine of $kx$, each having a different factor that depends on k and β; the factor of the sine must vanish, so that the solution can be glued symmetrically to its reflection. Since the factor is in general complex (hence its vanishing imposes two constraints, representing the two boundary conditions), this can in general be solved by adding an imaginary part of k, which gives the extra parameter needed. Thus E will have an imaginary part as well.

The physical meaning of this is that the standing wave in the middle decays; the waves newly emitted have therefore smaller amplitudes, so that their amplitude decays in time but grows with distance. The decay constant, denoted λ [1/s], is assumed small compared to $E/\backslash hbar$.

λ can be estimated without solving explicitly, by noting its effect on the probability current conservation law. Since the probability flows from the middle to the sides, we have:

: $\backslash frac\; \{\backslash partial\}\{\backslash partial\; t\}\; \backslash int\_\{-(q\_0+l)\}^\{(q\_0+l)\}\; \backslash Psi^*\backslash Psi\backslash \; dx\; =\; 2\; \backslash frac\{\backslash hbar\}\{2mi\}\backslash left(\backslash Psi\_1^*\; \backslash frac\{\backslash partial\; \backslash Psi\_1\; \}\{\backslash partial\; x\}-\; \backslash Psi\_1\; \backslash frac\{\backslash partial\; \backslash Psi\_1^*\; \}\{\backslash partial\; x\}\; \backslash right)\; ,$

note the factor of 2 is due to having two emitted waves.

Taking $\backslash Psi\backslash sim\; e^\{-\backslash lambda\; t\}$, this gives:

: $\backslash lambda\backslash frac\{2\}\{4\}\; (q\_0+l)\backslash left(A\backslash frac\{k\text{'}\}\{k\}\backslash right)^2\; e^\{2k\text{'}l\}\backslash approx2\backslash frac\{\backslash hbar\}\{m\}A^2k.$

Since the quadratic dependence on $k\text{'}l$ is negligible relative to its exponential dependence, we may write:

: $\backslash lambda\backslash approx4\backslash frac\{\backslash hbar\; k\}\{m(q\_0+l)\}\; \backslash frac\{k^2\}\{k\text{'}^2\}\backslash cdot\; e^\{-2k\text{'}l\}.$

Remembering the imaginary part added to k is much smaller than the real part, we may now neglect it and get:

: $\backslash lambda\backslash approx4\backslash frac\{\backslash hbar\; k\}\{m(q\_0+l)\}\backslash cdot\; \backslash frac\{E\}\{V-E\}\backslash cdot\; e^\{-2\backslash sqrt\{2m(V-E)\}l/\backslash hbar\}.$

Note that $\backslash frac\{\backslash hbar\; k\}\{m\}=\backslash sqrt\{2E/m\}$ is the particle velocity, so the first factor is the classical rate by which the particle trapped between the barriers ($2q\_0$ apart) hits them.

File:AlphaDecay.svg

Finally, moving to the three-dimensional problem, the spherically symmetric Schrödinger equation reads (expanding the wave function $\backslash psi(r,\backslash theta,\backslash phi)\; =\; \backslash chi(r)u(\backslash theta,\backslash phi)$ in spherical harmonics and looking at the l-th term):

: $\backslash frac\{\backslash hbar^2\}\{2m\}\backslash left(\backslash frac\{d^2\backslash chi\}\{dr^2\}+\backslash frac\{2\}\{r\}\backslash frac\{d\backslash chi\}\{dr\}\backslash right)=\backslash left(V(r)+\backslash frac\{\backslash hbar^2\}\{2m\}\backslash frac\{\backslash ell(\backslash ell+1)\}\{r^2\}-E\backslash right)\backslash chi.$

Since $\backslash ell>0$ amounts to enlarging the potential, and therefore substantially reducing the decay rate (given its exponential dependence on $\backslash sqrt\{V-E\}$): we focus on $\backslash ell=0$, and get a very similar problem to the previous one with $\backslash chi(r)\; =\; \backslash Psi(r)/r$, except that now the potential as a function of r is not a step function. In short $\backslash frac\{\backslash hbar^2\}\{2m\}\backslash left(\backslash ddot\backslash chi+\backslash frac\{2\}\{r\}\backslash dot\backslash chi\backslash right)=\backslash left(V(r)-E\backslash right)\backslash chi.$

The main effect of this on the amplitudes is that we must replace the argument in the exponent, taking an integral of $2\backslash sqrt\{2m(V-E)\}/\backslash hbar$ over the distance where $V(r)>E$ rather than multiplying by width l. We take the Coulomb potential:

: $V(r)\; =\; \backslash frac\; \{z(Z-z)\; e^2\}\{4\backslash pi\backslash varepsilon\_0\; r\}$

where $\backslash varepsilon\_0$ is the vacuum electric permittivity, e the electron charge, z = 2 is the charge number of the alpha particle and Z the charge number of the nucleus (Z–z after emitting the particle). The integration limits are then:

$r\_2\; =\; \backslash frac\; \{z(Z-z)\; e^2\}\{4\backslash pi\backslash varepsilon\_0\; E\},$ where we assume the nuclear potential energy is still relatively small, and

$r\_1$, which is where the nuclear negative potential energy is large enough so that the overall potential is smaller than E.

Thus, the argument of the exponent in λ is:

: $2\backslash frac\; \{\backslash sqrt\{2mE\}\}\{\backslash hbar\}\backslash int\_\{r\_1\}^\{r\_2\}\backslash sqrt\{\backslash frac\{V(r)\}\{E\}-1\}\; \backslash ,dr=2\backslash frac\{\backslash sqrt\{2mE\}\}\{\backslash hbar\}\backslash int\_\{r\_1\}^\{r\_2\}\backslash sqrt\{\backslash frac\{r\_2\}\{r\}-1\}\backslash ,dr.$

This can be solved by substituting $t=\backslash sqrt\{r/r\_2\}$ and then $t=\backslash cos(\backslash theta)$ and solving for θ, giving:

: $2r\_2\backslash frac\{\backslash sqrt\{2mE\}\}\{\backslash hbar\}[\backslash cos^\{-1\}(\backslash sqrt\{x\})-\backslash sqrt\{x\}\backslash sqrt\{1-x\}]=2\backslash frac\{\backslash sqrt\{2m\}z(Z-z)e^2\}\{4\backslash pi\backslash varepsilon\_0\backslash hbar\backslash sqrt\{E\}\}\backslash left[\backslash cos^\{-1\}(\backslash sqrt\{x\})-\backslash sqrt\{x\}\backslash sqrt\{1-x\}\backslash right]$

where $x\; =\; r\_1/r\_2$. Since x is small, the x-dependent factor is of the order 1.

Assuming $x\backslash ll\; 1$, the x-dependent factor can be replaced by $\backslash arccos0=\backslash pi/2,$ giving:

$\backslash lambda\backslash approx\; e^\{-\backslash sqrt\{\{E\_\{\backslash mathrm\{G\}\}\}/\{E\}\}\}$ with $E\_\{\backslash mathrm\; G\}=\backslash frac\{\backslash pi^2m/2\backslash left[z(Z-z)e^2\backslash right]^2\}\{(4\backslash pi\backslash varepsilon\_0\backslash hbar)^2\}.$

For a radium alpha decay, Z = 88, z = 2 and m ≈ 4m_p, E_G is approximately 50 GeV. Gamow calculated the slope of $\backslash log(\backslash lambda)$ with respect to E at an energy of 5 MeV to be ~ 10¹⁴ J⁻¹, compared to the experimental value of {{val|0.7|e=14|u=J⁻¹}}.{{efn|1=$\backslash log\backslash lambda\backslash approx-100$; $\{\backslash operatorname\{d\}\backslash log\backslash lambda\backslash over\backslash operatorname\{d\}\backslash !E\}=\backslash tfrac\{1\}\{2\}E\_\{\backslash rm\; G\}^\{1/2\}/E^\{3/2\}$ (Natural log.)

Around 10 /MeV; 'kT = 0.217 fJ = 0.135 keV', 'typical core temperatures in main-sequence stars (the Sun) give kT of the order of 1 keV': $2.17\backslash times10^\{-16\}$ joule}}

For an ideal gas, the Maxwell–Boltzmann distribution is proportional to

:$P\_\backslash text\{MB\}(E)\backslash propto\; e^\{-m\backslash langle\; v^2\backslash rangle/2k\_\{\backslash rm\; B\}T\}=e^\{-E/k\_\{\backslash rm\; B\}T\}$

where $\backslash langle\; v^2\; \backslash rangle$ is the average squared speed of all particles, $k\_\{\backslash rm\; B\}$ is the Boltzmann constant and T is absolute temperature.

The fusion probability is the product of the Maxwell–Boltzmann distribution factor and the Gamow factor

:$P\_\backslash text\{fusion\}(E)=P\_\backslash text\{MB\}(E)\backslash cdot\; P\_\backslash text\{G\}(E)\; =\backslash exp\backslash left(-\backslash frac\{E\}\{k\_\backslash mathrm\{B\}T\}-\backslash sqrt\{\backslash frac\{E\_\{\backslash rm\; G\}\}\{E\}\}\backslash right)$

The maximum of the fusion probability is given by $\backslash partial\; P\_\backslash text\{fusion\}/\backslash partial\; E=0,$ which yields{{Cite book |last=Clayton |first=D. D. (Donald Delbert) |url=https://archive.org/details/principlesofstel0000clay/mode/2up |title=Principles of stellar evolution and nucleosynthesis : with a new preface |date=1983 |publisher=Chicago ; London : University of Chicago Press |others=Internet Archive |isbn=978-0-226-10952-7}}

:$E\_\{\backslash rm\; max\}=\backslash left[E\_\{\backslash rm\; G\}\backslash left(\backslash frac\{k\_\{\backslash rm\; B\}T\}\{2\}\backslash right)^2\backslash right]^\{1/3\}.$

This quantity is known as the Gamow peak.{{efn|At kT resp. E_G the factors are 1/e (37%) at any temperature. Locus E₀ of the Gamow peak is $\backslash propto\; T^\{2/3\}.$}}

Expanding $P\_\backslash text\{fusion\}$ around $E\_\{\backslash rm\; max\}$ gives:

:$P\_\backslash text\{fusion\}(E)\backslash approx\; P\_\backslash text\{fusion\}(E\_\backslash text\{max\})\backslash cdot\backslash left[1+\backslash left(\backslash frac\{E-E\_\{\backslash rm\; max\}\}\{2\backslash Delta\}\backslash right)^2+\backslash cdots\backslash right],$

where (in joule)

:$\backslash Delta(T)=4\backslash sqrt\{\backslash frac\{E\_\{\backslash rm\; max\}k\_\{\backslash rm\; B\}T\}\{3\}\}=\backslash frac\{2^\{5/3\}\}\{\backslash sqrt\{3\}\}[E^\{\}\_\{\backslash rm\; G\}(k\_\{\backslash rm\; B\}T)^\{5\}]^\{1/6\}$

is the Gamow window.{{efn|Double log. graphs $\backslash Delta$ vs T: $2\backslash log(\backslash Delta/4)=\backslash log(k\_\{\backslash rm\; B\}T)+\backslash log(E\_\{\backslash rm\; max\}/3)=[5\backslash log(k\_\{\backslash rm\; B\}T)+\backslash log(E\_\{\backslash rm\; G\})-3\backslash log(3)-2\backslash log(2)]/3$. Similar $\backslash log(\backslash Delta/4)=[5\backslash log(k\_\{\backslash rm\; B\}T)+\backslash log(E\_\{\backslash rm\; G\})-\backslash log(27\backslash cdot4)]/6$}}

In 1927, Ernest Rutherford published an article in Philosophical Magazine on a problem related to Hans Geiger's 1921 experiment of scattering alpha particles from uranium.{{Cite journal |last=Merzbacher |first=Eugen |date=2002-08-01 |title=The Early History of Quantum Tunneling |url=https://pubs.aip.org/physicstoday/article-abstract/55/8/44/412308/The-Early-History-of-Quantum-Tunneling-Molecular?redirectedFrom=fulltext |journal=Physics Today |volume=55 |issue=8 |pages=44–49 |doi=10.1063/1.1510281 |issn=0031-9228|url-access=subscription }} Previous experiments with thorium C' (now called polonium-262){{efn|₈₈Ra (₉₀Th) ₉₂U are radioactive elements in periods 7, ₈₄Po in period 6. Spontaneous nuclear fission}} confirmed that uranium has a Coulomb barrier of 8.57 MeV, however uranium emitted alpha particles of 4.2 MeV. The emitted energy was too low to overcome the barrier. On 29 July 1928, George Gamow, and independently the next day Ronald Wilfred Gurney and Edward Condon submitted their solution based on quantum tunnelling to the journal Zeitschrift für Physik. Their work was based on previous work on tunnelling by J. Robert Oppenheimer, Gregor Wentzel, Lothar Wolfgang Nordheim, and Ralph H. Fowler. Gurney and Condon cited also Friedrich Hund.

In 1931, Arnold Sommerfeld introduced a similar factor (a Gaunt factor) for the discussion of bremsstrahlung.{{Cite book |last=Iben |first=Icko |url=https://www.google.fr/books/edition/Stellar_Evolution_Physics/hFmpIXwLUvIC?hl=en&gbpv=1&dq=sommerfeld++1931+factor&pg=PA335&printsec=frontcover |title=Stellar Evolution Physics |date=2013 |publisher=Cambridge University Press |isbn=978-1-107-01656-9 |language=en}}

Gamow popularized his personal version of the discovery in his 1970's book, My World Line: An Informal Autobiography.

• Stellar_nucleosynthesis#Reaction_rate

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• [https://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/alpdec.html Modeling Alpha Half-life (Georgia State University)] hyperphysics.phy-astr.gsu.edu

Category:Nuclear physics

Category:George Gamow