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  • introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 1

    9 – Nuclear reaction rates

    introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 1

  • introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 2

    9.1 – Nuclear decays: Neutron decay By increasing the neutron number, the gain in binding energy of a nucleus due to the volume term is exceeded by the loss due to the growing asymmetry term. After that, no more neutrons can be bound, the neutron drip line is reached beyond which, neutron decay occurs:

    Neutron drip line: Sn = 0

    (Z, N) à (Z, N - 1) + n

    Q-value: Qn = m(Z, N) - m(Z, N-1) - mn

    Beyond the drip line Sn < 0

    Neutron Separation Energy Sn Sn(Z, N) = m(Z, N-1) + mn - m(Z, N) = -Qn for n-decay

    neutron unbound

    (9.1) (9.2)

    (9.3)

    Gamma-decay •  When a reaction places a nucleus in an excited state, it drops to the lowest

    energy through gamma emission •  Excited states and decays in nuclei are similar to atoms: (a) they conserve

    angular momentum and parity, (b) the photon has spin = 1 and parity = -1 •  For orbital with angular momentum L, parity is given by P= (-1)L •  To first order, the photon carries away the energy from an electric dipole

    transition in nuclei. But in nuclei it is easier to have higher order terms in than in atoms.

  • introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 3

    Proton decay

    Proton drip line: Sp = 0

    beyond the drip line: Sp < 0

    Proton Separation Energy Sp: Sp(Z,N) = m(Z-1,N) + mp - m(Z,N)

    the nuclei are proton unbound

    α-decay is the emission of an a particle ( = 4He nucleus)

    Qα = m(Z,A)−m(Z− 2,A− 4)−mα"# $%c 2The Q-value for a decay is given by

    Nuclear lifetimes for decay through all the modes described so far depend not only on the Q-values, but also on the conservation of other physical quantities, such as angular momentum. Typical decay times for similar lifetimes are (a) 10-10 s for beta decay, (b) 10-16 s for gamma-decay and (c) up to many years for alpha- decay.

    α-decay

    Decay lifetimes

    (9.4)

    (9.5)

  • introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 4 4

    Fission Very heavy nuclei can fission into two parts (Q > 0) This happens because for large nuclei the surface energy is less important – large deformations are then possible. With a small amount of additional energy the nucleus can be deformed sufficiently so that Coulomb repulsion wins over nucleon-nucleon attraction and the nucleus fissions. This will be possible if

    Qf = m(Z,A)− 2m(Z / 2,A / 2)"# $%c 2 > 0

    Using the LDM formula, Eq. (7.5) we can shown that Qα > 0 this happens for A > 90.

    Nuc le a r C ha rg e Yie ld in Fissio n o f 234U

    2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5

    8 0 1 0 0 1 2 0 1 4 0 1 6 0 Ma ss Num b e r A

    Pro to n Num b e r Z

    0

    5

    1 0

    1 5

    2 0

    Yi el d Y ( Z)

    (% )

    Example from Moller et al., Nature 409 (2001) 485

    Symmetric fission (fission into two nearly equal size fragments) as well as asymmetric fission are likely to occur.

    (9.6)

  • introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 5

    1.  Most nuclei found in nature are the stable and not too heavy to undergo alpha decay or fission)

    2. There are many more unstable nuclei that can exist for short times after their creation. 3. Alpha particle is favored building blocks of nuclei because of their comparably low Coulomb barrier when fused with other nuclei high binding energy per nucleon 4. The binding energy per nucleon has a maximum in the Iron-Nickel region

    This explains abundance peaks at nuclei composed of multiples of alpha particles in solar system abundance distribution

    9.2 - Observations

    The iron peak in solar abundance distribution implies that some fraction of matter was brought into equilibrium, probably during supernovae explosions.

  • introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 6

    Observations – Solar abundances

  • introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 7

    9.3 - Nuclear reactions in stars The plasma within a star is a mix of fully ionized projectiles and target nuclei at a temperature T. Using the definition of cross section in Eq. (7.24), we get that for particles with a given relative velocity v and within a volume V with projectile number density ni, and target number density nj:

    λ =σ niv R =σ nivn jV

    so for reaction rate per second and cm3 is r = nin jσ v

    This is proportional to the number of i-j pairs in the volume. However, when the species are identical, i.i., i =j, one has to divide by 2 to avoid double counting

    r = 1 1+δij

    nin jσ v

    (9.7)

    (9.8)

    (9.9)

  • introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 8

    Reaction rates at temperature T Assuming Local Thermal Equilibrium LTE the velocity distribution of particles in a plasma follow a Maxwell-Boltzmann velocity distribution:

    where Φ(v) is the probability to find a particle with a velocity between v and v + dv.

    kT2 v

    2 2/3 2

    e 2

    4)( m

    v kT mv

    ⎟⎟ ⎠

    ⎞ ⎜⎜ ⎝

    ⎛ =Φ

    π π ∫ =Φ 1)( dvvwith

    max at E = kT

    This distribution is easily deduced from the Boltzmann distribution f(E) = A exp(-E/kT) by replacing E = mv2/2 and adjusting the constant A to conform with the normalization condition given by Eq. (9.11).

    (9.10) (9.11)

  • introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 9

    r = 1 1+δij

    nin j σ (v)Φ(v)vdv∫

    kT2 v

    2 2/3 2

    e 2

    4)( µ

    π µ

    π −

    ⎟⎟ ⎠

    ⎞ ⎜⎜ ⎝

    ⎛ =Φ v

    kT v

    In terms of the Maxwell-Boltzmann distribution, with the mass m replaced by the reduced mass m of the i and j particles, i.e.,

    µ = mimj mi +m j

    In the stellar environments, the reaction rates given in Eq. (9.9) has to be averaged over velocities,

    r = 1 1+δij

    nin j

    using over the distribution Φ(v),

    Reaction rates at temperature T

    (9.12) (9.13)

    (9.14)

    (9.15)

  • introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 10

    λ = 1

    1+δij YiρNA

    r = 1 1+δij

    YiYjρ 2 NA

    2

    Sometimes, one uses another notation for reactions per second and target nucleus j, by dividing the above equation by YjρNA, i.e.,

    One usually call this by the stellar reaction rate of a specific reaction

    In terms of units of stellar reaction rate given by NA in cm3/s/g, the reaction rate per second and per cm3, is

    Reaction rates in terms of abundances

    (9.16)

    (9.17)

    As an application, lets assume that the species i and j are destructed by j capturing the projectile i in the form i + j à k. Then

    dn j dt

    = −n jλ = −n jYiρNA

    dnk dt

    = nkλ

    (9.18)

  • introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 11

    Since the dnj/dt is proportional to –nj, the solution of the equation is a decaying exponential. And since the species k is built from the destruction of of j, its time dependence should be proportional to [1 – nj(t)]. In other words,

    n j(t) = n0 j e −λ t

    nk (t) = n0k (1− e −λ t )

    Reaction rates in terms of abundances

    (9.19)

    In terms of the abundances: Yj(t) =Y0j e

    −λ t

    Yk (t) =Y0k (1− e −λ t )

    (9.20)

    The lifetime for the destruction of j is

    τ = 1 λ =

    1 YjρNA

    (9.21)

    After a time equal to a few multiples of τ, the species j is almost completely destroyed while k is almost completely built up from i and j capture. One often uses the name “half-life”, τ1/2, instead of lifetime, where τ1/2 = τ ln2.

  • introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 12

    9.4 - Energy generation Lets consider the energy generation by nuclear reaction sin stars. As we know the energy released is given by the Q-values

    ⎟⎟ ⎠

    ⎞ ⎜⎜ ⎝

    ⎛ −= ∑ ∑

    i nuclei initial j nuclei final

    2 ji mmcQ

    where m are the respective masses of the particles. The energy generated per gram and second by a reaction i + j is then

    ε = rQ ρ =Q 1 1+δij

    YiYjρNA 2

    We can now define the reaction flow as the abundance of nuclei converted in time T from species j to k via a specific reaction as

    F = dt dYj dt

    !

    " ##

    $

    % &&

    0

    T

    ∫ via specific reaction

    = λ(t)Yj(t)dt 0

    T

    (9.22)

    (9.23)

    (9.24)

  • introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 13

    Branching Consider possibilities as in the figure on the side in terms of several possible reaction rates λi

    The total destruction rate for the nucleus

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