Neutron scattering length

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A neutron may pass by a nucleus with a probability determined by the nuclear interaction distance, or be absorbed, or undergo scattering that may be either coherent or incoherent. [1] The interference effects in coherent scattering can be computed via the coherent scattering length of neutrons, being proportional to the amplitude of the spherical scattered waves according to Huygens–Fresnel theory. This scattering length varies by isotope (and by element as the weighted arithmetic mean over the constituent isotopes) in a way that appears random, whereas the X-ray scattering length is just the product of atomic number and Thomson scattering length, thus monotonically increasing with atomic number. [1] [2]

The scattering length may be either positive or negative. The scattering cross-section is equal to the square of the scattering length multiplied by 4π, [3] i.e. the area of a circle with radius twice the scattering length. In some cases, as with titanium and nickel, it is possible to mix isotopes of an element whose lengths are of opposite signs to give a net scattering length of zero, in which case coherent scattering will not occur at all, while for vanadium already the opposite signs of the only naturally occurring isotope's two spin configurations give a near cancellation. However, neutrons will still undergo strong incoherent scattering in these materials. [1]

There is a large difference in scattering length between protium (-0.374) and deuterium (0.667). By using heavy water as solvent and/or selective deuteration of the probed molecule (exchanging the naturally occurring protium by deuterium) this difference can be leveraged in order to image the hydrogen configuration in organic matter, which is nearly impossible with X-rays due to their small sensitivity to hydrogen's single electron. [4] On the other hand, neutron scattering studies of hydrogen-containing samples often suffer from the strong incoherent scattering of natural hydrogen.

element protons isotopeneutron scattering length
bcoh (fm)
coherent
cross-section
σcoh (barn)
incoherent
cross-section
σinc (barn)
absorption
cross-section
σa (barn)
Hydrogen 11 [2] [5] -3.74 [1] [2] [5] [6] 1.758 [1] 79.7, [6] 80.27 [1] 0.33, [6] 0.383 [1]
Hydrogen 126.67 [1] [2] [5] [6] 5.592 [1] 2.0, [6] 2.05 [1] 0.0005 [1] [6]
Boron 5natural5.30 [1] 3.54 [1] 1.70 [1] 767.0 [1]
Carbon 6126.65 [1] [2] [5] [6] 5.550 [1] 0.0, [6] 0.001 [1] 0.0035, [6] 0.004 [1]
Nitrogen 7149.36, [1] 9.40, [2] 9.4 [5] [6] 11.01 [1] 0.3, [6] 0.5 [1] 1.9 [1] [6]
Oxygen 8165.80, [2] 5.8 [1] [5] [6] 4.232 [1] 0.0, [6] 0.000 [1] 0.00019, [6] 0.0002 [1]
Aluminum 13natural3.45, [1] 3.5 [6] 1.495 [1] 0.0, [6] 0.008 [1] 0.23, [6] 0.231 [1]
Silicon 14natural4.2 [6] [7] 0.0 [6] 0.17 [6]
Phosphorus 15305.10 [2]
Sulfur 16322.80, [2] 2.8 [5]
Titanium 22natural-3.44, [1] -3.4 [6] [7] 1.485 [1] 2.87, [1] 3.0 [6] 6.09, [1] 6.1 [6]
Vanadium 23natural-0.38 [1] 0.018 [1] 5.07 [1] 5.08 [1]
Chromium 24natural3.64 [1] 1.66 [1] 1.83 [1] 3.05 [1]
Manganese 2555 (natural)-3.73 [1] 1.75 [1] 0.4 [1] 13.3 [1]
Iron 26natural9.45, [1] 9.5 [6] 11.22 [1] 0.4 [1] [6] 2.56, [1] 2.6 [6]
Nickel 28natural10.3 [1] 13.3 [1] 5.2 [1] 4.49 [1]
Copper 29natural7.72 [1] 7.485 [1] 0.55 [1] 3.78 [1]
Zirconium 40natural7.16, [1] 0.72 [6] 6.44 [1] 0.02, [1] 0.3 [6] 0.18, [6] 0.185 [1]
Niobium 4193 (natural)7.054 [1] 6.253 [1] 0.0024 [1] 1.15 [1]
Molybdenum 42natural6.72 [1] 5.67 [1] 0.04 [1] 2.48 [1]
Cadmium 48natural4.87 [1] 3.04 [1] 3.46 [1] 2520 [1]
Tin 50natural6.23 [1] 4.87 [1] 0.022 [1] 0.626 [1]
Cerium 58natural4.8 [6] 0.0 [6] 0.63 [6]
Gadolinium 64natural6.5 [1] 29.3 [1] 151 [1] 49700 [1]
Tantalum 73natural6.91 [1] 6.00 [1] 0.01 [1] 20.6 [1]
Tungsten 74natural4.86 [1] 2.97 [1] 1.63 [1] 18.3 [1]
Gold 791977.60 [2]
Lead 82natural9.41 [1] 11.115 [1] 0.003 [1] 0.171 [1]
Thorium 90232 (natural)9.8 [6] 0.00 [6] 7.4 [6]
Uranium 92natural8.42 [1] [6] 8.903 [1] 0.00, [6] 0.005 [1] 7.5, [6] 7.57 [1]

More comprehensive data is available from NIST [8] and Atominstitut of Vienna. [9]

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A chemical element is a chemical substance that cannot be broken down into other substances by chemical reactions. The basic particle that constitutes a chemical element is the atom. Chemical elements are identified by the number of protons in the nuclei of their atoms, known as the element's atomic number. For example, oxygen has an atomic number of 8, meaning that each oxygen atom has 8 protons in its nucleus. Two or more atoms of the same element can combine to form molecules, in contrast to chemical compounds or mixtures, which contain atoms of different elements. Atoms can be transformed into different elements in nuclear reactions, which change an atom's atomic number.

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H
or D, also known as heavy hydrogen) is one of two stable isotopes of hydrogen (the other being protium, or hydrogen-1). The nucleus of a deuterium atom, called a deuteron, contains one proton and one neutron, whereas the far more common protium has no neutrons in the nucleus. Deuterium has a natural abundance in Earth's oceans of about one atom of deuterium among every 6,420 atoms of hydrogen (see heavy water). Thus deuterium accounts for approximately 0.0156% by number (0.0312% by mass) of all the naturally occurring hydrogen in the oceans (i.e., 4.85×1013 tonnes of deuterium – mainly in form of HOD and only rarely in form of D2O – in 1.4×1018 tonnes of water), while protium accounts for 99.98%. The abundance of deuterium changes slightly from one kind of natural water to another (see Vienna Standard Mean Ocean Water)

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<span class="mw-page-title-main">Neutron</span> Subatomic particle with no charge

The neutron is a subatomic particle, symbol
n
or
n0
, which has a neutral charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behave similarly within the nucleus, they are both referred to as nucleons. Nucleons have a mass of approximately one atomic mass unit, or dalton, symbol Da. Their properties and interactions are described by nuclear physics. Protons and neutrons are not elementary particles; each is composed of three quarks.

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Hydrogen (1H) has three naturally occurring isotopes, sometimes denoted 1
H
, 2
H
, and 3
H
. 1
H
and 2
H
are stable, while 3
H
has a half-life of 12.32(2) years. Heavier isotopes also exist, all of which are synthetic and have a half-life of less than one zeptosecond (10−21 s). Of these, 5
H
is the least stable, while 7
H
is the most.

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