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Quantum Numbers and Rules

Module by: OpenStax College. E-mail the author

Summary:

  • Define quantum number.
  • Calculate angle of angular momentum vector with an axis.
  • Define spin quantum number.

Physical characteristics that are quantized—such as energy, charge, and angular momentum—are of such importance that names and symbols are given to them. The values of quantized entities are expressed in terms of quantum numbers, and the rules governing them are of the utmost importance in determining what nature is and does. This section covers some of the more important quantum numbers and rules—all of which apply in chemistry, material science, and far beyond the realm of atomic physics, where they were first discovered. Once again, we see how physics makes discoveries which enable other fields to grow.

The energy states of bound systems are quantized, because the particle wavelength can fit into the bounds of the system in only certain ways. This was elaborated for the hydrogen atom, for which the allowed energies are expressed as En1/n2En1/n2, where n=1, 2, 3, ...n=1, 2, 3, .... We define nn to be the principal quantum number that labels the basic states of a system. The lowest-energy state has n=1n=1, the first excited state has n=2n=2, and so on. Thus the allowed values for the principal quantum number are

n=1, 2, 3, ....n=1, 2, 3, .... size 12{n=1, 2, 3, "." "." "." } {}
(1)

This is more than just a numbering scheme, since the energy of the system, such as the hydrogen atom, can be expressed as some function of nn size 12{n} {}, as can other characteristics (such as the orbital radii of the hydrogen atom).

The fact that the magnitude of angular momentum is quantized was first recognized by Bohr in relation to the hydrogen atom; it is now known to be true in general. With the development of quantum mechanics, it was found that the magnitude of angular momentum LL size 12{L} {} can have only the values

L=ll+1hl=0, 1, 2, ...,n1,L=ll+1h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {}l=0, 1, 2, ...,n1, size 12{ left (l=0, 1, 2, "." "." "." ,n - 1 right )} {}
(2)

where ll size 12{l} {} is defined to be the angular momentum quantum number. The rule for ll size 12{l} {} in atoms is given in the parentheses. Given nn size 12{n} {}, the value of ll size 12{l} {} can be any integer from zero up to n1n1 size 12{n - 1} {}. For example, if n=4n=4 size 12{n=4} {}, then ll size 12{l} {} can be 0, 1, 2, or 3.

Note that for n=1n=1 size 12{n=1} {}, ll size 12{l} {} can only be zero. This means that the ground-state angular momentum for hydrogen is actually zero, not h/2π h/2π as Bohr proposed. The picture of circular orbits is not valid, because there would be angular momentum for any circular orbit. A more valid picture is the cloud of probability shown for the ground state of hydrogen in (Reference). The electron actually spends time in and near the nucleus. The reason the electron does not remain in the nucleus is related to Heisenberg’s uncertainty principle—the electron’s energy would have to be much too large to be confined to the small space of the nucleus. Now the first excited state of hydrogen has n=2n=2 size 12{n=2} {}, so that ll size 12{l} {} can be either 0 or 1, according to the rule in L=ll+1hL=ll+1h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} . Similarly, for n=3n=3 size 12{n=3} {}, ll size 12{l} {} can be 0, 1, or 2. It is often most convenient to state the value of ll size 12{l} {}, a simple integer, rather than calculating the value of LL size 12{L} {} from L=ll+1hL=ll+1h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {}. For example, for l=2l=2 size 12{l=2} {}, we see that

L=22+1h=6h=0.390h=2.58×1034 Js.L=22+1h=6h=0.390h=2.58×1034 Js. size 12{L= sqrt {2 left (2+1 right )} { {h} over {2π} } = sqrt {6} { {h} over {2π} } =0 "." "390"h=2 "." "58" times "10" rSup { size 8{ - "34"} } " J" cdot s} {}
(3)

It is much simpler to state l=2l=2 size 12{l=2} {}.

As recognized in the Zeeman effect, the direction of angular momentum is quantized. We now know this is true in all circumstances. It is found that the component of angular momentum along one direction in space, usually called the zz size 12{z} {}-axis, can have only certain values of LzLz size 12{L rSub { size 8{z} } } {}. The direction in space must be related to something physical, such as the direction of the magnetic field at that location. This is an aspect of relativity. Direction has no meaning if there is nothing that varies with direction, as does magnetic force. The allowed values of LzLz size 12{L rSub { size 8{z} } } {} are

Lz=mlhml=l,l+1, ...,1, 0, 1, ...l1,l,Lz=mlh size 12{L rSub { size 8{z} } =m rSub { size 8{l} } { {h} over {2π} } } {}ml=l,l+1, ...,1, 0, 1, ...l1,l, size 12{ left (m rSub { size 8{l} } = - l, - l+1, "." "." "." , - 1, 0, 1, "." "." "." l - 1, l right )} {}
(4)

where LzLz size 12{L rSub { size 8{z} } } {} is the zz size 12{z} {}-component of the angular momentum and mlml size 12{m rSub { size 8{l} } } {} is the angular momentum projection quantum number. The rule in parentheses for the values of mlml size 12{m rSub { size 8{l} } } {} is that it can range from ll size 12{ - l} {} to ll size 12{l} {} in steps of one. For example, if l=2l=2 size 12{l=2} {}, then mlml size 12{m rSub { size 8{l} } } {} can have the five values –2, –1, 0, 1, and 2. Each mlml size 12{m rSub { size 8{l} } } {} corresponds to a different energy in the presence of a magnetic field, so that they are related to the splitting of spectral lines into discrete parts, as discussed in the preceding section. If the zz size 12{z} {}-component of angular momentum can have only certain values, then the angular momentum can have only certain directions, as illustrated in Figure 1.

Figure 1: The component of a given angular momentum along the zz-axis (defined by the direction of a magnetic field) can have only certain values; these are shown here for l=1l=1, for which ml=1, 0, and +1ml=1, 0, and +1. The direction of LL is quantized in the sense that it can have only certain angles relative to the zz-axis.
The image shows two possible values of component of a given angular momentum along z-axis. One circular orbit above the original circular orbit is shown for m sub l value of plus one. Another circular orbit below the original circular orbit is shown for m sub l value of minus one. The angular momentum vector for the top circular orbit makes an angle of theta sub one with the vertical axis. The horizontal angular momentum vector at original circular orbit makes an angle of theta sub two with the vertical axis. The angular momentum vector for the bottom circular orbit makes an angle of theta sub three with the vertical axis.

Example 1: What Are the Allowed Directions?

Calculate the angles that the angular momentum vector LL size 12{L} {} can make with the zz size 12{z} {}-axis for l=1l=1 size 12{l=1} {}, as illustrated in Figure 1.

Strategy

Figure 1 represents the vectors LL size 12{L} {} and LzLz size 12{L rSub { size 8{z} } } {} as usual, with arrows proportional to their magnitudes and pointing in the correct directions. LL size 12{L} {} and LzLz size 12{L rSub { size 8{z} } } {} form a right triangle, with LL size 12{L} {} being the hypotenuse and LzLz size 12{L rSub { size 8{z} } } {} the adjacent side. This means that the ratio of LzLz size 12{L rSub { size 8{z} } } {} to LL size 12{L} {} is the cosine of the angle of interest. We can find LL size 12{L} {} and LzLz size 12{L rSub { size 8{z} } } {} using L=ll+1hL=ll+1h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} and Lz=mhLz=mh size 12{L rSub { size 8{z} } =m { {h} over {2π} } } {}.

Solution

We are given l=1l=1 size 12{l=1} {}, so that mlml size 12{m rSub { size 8{l} } } {} can be +1, 0, or −1. Thus LL size 12{L} {} has the value given by L=ll+1hL=ll+1h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {}.

L=ll+1h=2hL=ll+1h=2h size 12{L= { { sqrt {l left (l+1 right )} h} over {2π} } = { { sqrt {2} h} over {2π} } } {}
(5)

LzLz size 12{L rSub { size 8{z} } } {} can have three values, given by Lz=mlhLz=mlh size 12{L rSub { size 8{z} } =m rSub { size 8{l} } { {h} over {2π} } } {}.

Lz=mlh={ h, ml = +1 0, ml = 0 h, ml = 1 Lz=mlh={ h, ml = +1 0, ml = 0 h, ml = 1
(6)

As can be seen in Figure 1, cos θ= L z /L, cos θ= L z /L, and so for ml=+1ml=+1 size 12{m rSub { size 8{l} } "=+"1} {}, we have

cosθ1=LZL=h2h=12=0.707.cosθ1=LZL=h2h=12=0.707. size 12{"cos"θ rSub { size 8{1} } = { {L rSub { size 8{Z} } } over {L} } = { { { {h} over {2π} } } over { { { sqrt {2} h} over {2π} } } } = { {1} over { sqrt {2} } } =0 "." "707"} {}
(7)

Thus,

θ1=cos10.707=45.0º.θ1=cos10.707=45.0º.
(8)

Similarly, for ml=0ml=0 size 12{m rSub { size 8{l} } =0} {}, we find cosθ2=0cosθ2=0 size 12{"cos"θ rSub { size 8{2} } =0} {}; thus,

θ2=cos10=90.0º.θ2=cos10=90.0º. size 12{θ rSub { size 8{2} } ="cos" rSup { size 8{ - 1} } 0="90" "." 0°} {}
(9)

And for ml=1ml=1 size 12{m rSub { size 8{l} } = - 1} {},

cosθ3=LZL=h2h=12=0.707,cosθ3=LZL=h2h=12=0.707, size 12{"cos"θ rSub { size 8{3} } = { {L rSub { size 8{Z} } } over {L} } = { { - { {h} over {2π} } } over { { { sqrt {2} h} over {2π} } } } = - { {1} over { sqrt {2} } } = - 0 "." "707"} {}
(10)

so that

θ3=cos10.707=135.0º.θ3=cos10.707=135.0º. size 12{θ rSub { size 8{3} } ="cos" rSup { size 8{ - 1} } left ( - 0 "." "707" right )="135" "." 0°} {}
(11)

Discussion

The angles are consistent with the figure. Only the angle relative to the zz size 12{z} {}-axis is quantized. LL size 12{L} {} can point in any direction as long as it makes the proper angle with the zz size 12{z} {}-axis. Thus the angular momentum vectors lie on cones as illustrated. This behavior is not observed on the large scale. To see how the correspondence principle holds here, consider that the smallest angle ( θ 1 θ 1 in the example) is for the maximum value of ml=0ml=0, namely ml=lml=l. For that smallest angle,

cosθ=LzL=lll+1,cosθ=LzL=lll+1, size 12{"cos"θ= { {L rSub { size 8{z} } } over {L} } = { {l} over { sqrt {l left (l+1 right )} } } } {}
(12)

which approaches 1 as ll size 12{l} {} becomes very large. If cosθ=1cosθ=1 size 12{"cos"θ=1} {}, then θ=θ=. Furthermore, for large ll, there are many values of mlml, so that all angles become possible as ll gets very large.

Intrinsic Spin Angular Momentum Is Quantized in Magnitude and Direction

There are two more quantum numbers of immediate concern. Both were first discovered for electrons in conjunction with fine structure in atomic spectra. It is now well established that electrons and other fundamental particles have intrinsic spin, roughly analogous to a planet spinning on its axis. This spin is a fundamental characteristic of particles, and only one magnitude of intrinsic spin is allowed for a given type of particle. Intrinsic angular momentum is quantized independently of orbital angular momentum. Additionally, the direction of the spin is also quantized. It has been found that the magnitude of the intrinsic (internal) spin angular momentum, SS size 12{S} {}, of an electron is given by

S=ss+1h(s=1/2 for electrons),S=ss+1h(s=1/2 for electrons), size 12{s=1/2} {}
(13)

where ss size 12{s} {} is defined to be the spin quantum number. This is very similar to the quantization of LL size 12{L} {} given in L=ll+1hL=ll+1h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {}, except that the only value allowed for ss size 12{s} {} for electrons is 1/2.

The direction of intrinsic spin is quantized, just as is the direction of orbital angular momentum. The direction of spin angular momentum along one direction in space, again called the zz size 12{z} {}-axis, can have only the values

S z = m s h m s = 1 2 , + 1 2 S z = m s h size 12{S rSub { size 8{z} } =m rSub { size 8{s} } { {h} over {2π} } } {} m s = 1 2 , + 1 2 size 12{ left (m rSub { size 8{s} } = - { {1} over {2} } , + { {1} over {2} } right )} {}
(14)

for electrons. SzSz size 12{S rSub { size 8{z} } } {} is the zz size 12{z} {}-component of spin angular momentum and msms size 12{S rSub { size 8{z} } } {} is the spin projection quantum number. For electrons, ss size 12{s} {} can only be 1/2, and msms size 12{m rSub { size 8{s} } } {} can be either +1/2 or –1/2. Spin projection ms=+1/2ms=+1/2 size 12{m rSub { size 8{s} } "=+"1/2} {} is referred to as spin up, whereas ms=1/2ms=1/2 size 12{m rSub { size 8{s} } = - 1/2} {} is called spin down. These are illustrated in (Reference).

Intrinsic Spin:

In later chapters, we will see that intrinsic spin is a characteristic of all subatomic particles. For some particles ss size 12{s} {} is half-integral, whereas for others ss size 12{s} {} is integral—there are crucial differences between half-integral spin particles and integral spin particles. Protons and neutrons, like electrons, have s=1/2s=1/2 size 12{s=1/2} {}, whereas photons have s=1s=1 size 12{s=1} {}, and other particles called pions have s=0s=0 size 12{s=0} {}, and so on.

To summarize, the state of a system, such as the precise nature of an electron in an atom, is determined by its particular quantum numbers. These are expressed in the form n, l,ml,msn, l,ml,ms —see Table 1 For electrons in atoms, the principal quantum number can have the values n=1, 2, 3, ...n=1, 2, 3, .... Once nn is known, the values of the angular momentum quantum number are limited to l=1, 2, 3, ...,n1l=1, 2, 3, ...,n1. For a given value of ll, the angular momentum projection quantum number can have only the values ml=l,l+1, ...,1, 0, 1, ...,l1,lml=l,l+1, ...,1, 0, 1, ...,l1,l. Electron spin is independent of n, l,n, l, and mlml, always having s=1/2s=1/2. The spin projection quantum number can have two values, ms=1/2 or 1/2ms=1/2 or 1/2.

Table 1: Atomic Quantum Numbers
Name Symbol Allowed values
Principal quantum number n n 1, 2, 3, ... 1, 2, 3, ...
Angular momentum l l 0, 1, 2, ... n 1 0, 1, 2, ... n 1
Angular momentum projection m l m l l , l + 1, ... , 1, 0, 1, ... , l 1, l ( or 0, ±1, ±2, ... , ± l ) l , l + 1, ... , 1, 0, 1, ... , l 1, l ( or 0, ±1, ±2, ... , ± l )
Spin1 s s 1/2 ( electrons ) 1/2 ( electrons )
Spin projection m s m s 1/2, + 1/2 1/2, + 1/2

Figure 2 shows several hydrogen states corresponding to different sets of quantum numbers. Note that these clouds of probability are the locations of electrons as determined by making repeated measurements—each measurement finds the electron in a definite location, with a greater chance of finding the electron in some places rather than others. With repeated measurements, the pattern of probability shown in the figure emerges. The clouds of probability do not look like nor do they correspond to classical orbits. The uncertainty principle actually prevents us and nature from knowing how the electron gets from one place to another, and so an orbit really does not exist as such. Nature on a small scale is again much different from that on the large scale.

Figure 2: Probability clouds for the electron in the ground state and several excited states of hydrogen. The nature of these states is determined by their sets of quantum numbers, here given as n,l,mln,l,ml size 12{ left (n, l, m rSub { size 8{l} } right )} {}. The ground state is (0, 0, 0); one of the possibilities for the second excited state is (3, 2, 1). The probability of finding the electron is indicated by the shade of color; the darker the coloring the greater the chance of finding the electron.
The image shows probability clouds for the electron in the ground state and several excited states of hydrogen. Sets of quantum numbers given as n l m subscript l are shown for each state. The ground state is zero zero zero. The probability of finding the electron is indicated by the shade of color.

We will see that the quantum numbers discussed in this section are valid for a broad range of particles and other systems, such as nuclei. Some quantum numbers, such as intrinsic spin, are related to fundamental classifications of subatomic particles, and they obey laws that will give us further insight into the substructure of matter and its interactions.

PhET Explorations: Stern-Gerlach Experiment:

The classic Stern-Gerlach Experiment shows that atoms have a property called spin. Spin is a kind of intrinsic angular momentum, which has no classical counterpart. When the z-component of the spin is measured, one always gets one of two values: spin up or spin down.

Section Summary

  • Quantum numbers are used to express the allowed values of quantized entities. The principal quantum number nn size 12{n} {} labels the basic states of a system and is given by
    n=1,2,3,....n=1,2,3,.... size 12{n=1, 2, 3, "." "." "." } {}
    (15)
  • The magnitude of angular momentum is given by
    L = l l + 1 h l = 0, 1, 2, ... , n 1 , L = l l + 1 h l = 0, 1, 2, ... , n 1 ,
    (16)
    where ll size 12{l} {} is the angular momentum quantum number. The direction of angular momentum is quantized, in that its component along an axis defined by a magnetic field, called the zz size 12{z} {}-axis is given by
    Lz=mlhml=l,l+1, ...,1, 0, 1, ...l1,l,Lz=mlh size 12{L rSub { size 8{z} } =m rSub { size 8{l} } { {h} over {2π} } } {}ml=l,l+1, ...,1, 0, 1, ...l1,l,
    (17)
    where LzLz size 12{L rSub { size 8{z} } } {} is the zz size 12{z} {}-component of the angular momentum and mlml size 12{m rSub { size 8{l} } } {} is the angular momentum projection quantum number. Similarly, the electron’s intrinsic spin angular momentum SS size 12{S} {} is given by
    S=ss+1h(s=1/2 for electrons),S=ss+1h( size 12{S= sqrt {s left (s+1 right )} { {h} over {2π} } } {}s=1/2 for electrons), size 12{s=1/2} {}
    (18)
    ss size 12{s} {} is defined to be the spin quantum number. Finally, the direction of the electron’s spin along the zz size 12{z} {}-axis is given by
    Sz=mshms=12,+12,Sz=msh size 12{S rSub { size 8{z} } =m rSub { size 8{s} } { {h} over {2π} } } {}ms=12,+12, size 12{ left (m rSub { size 8{s} } = - { {1} over {2} } , + { {1} over {2} } right )} {}
    (19)
    where SzSz size 12{S rSub { size 8{z} } } {} is the zz size 12{z} {}-component of spin angular momentum and msms size 12{m rSub { size 8{s} } } {} is the spin projection quantum number. Spin projection ms=+1/2ms=+1/2 size 12{m rSub { size 8{s} } "=+"1/2} {} is referred to as spin up, whereas ms=1/2ms=1/2 size 12{m rSub { size 8{s} } = - 1/2} {} is called spin down. Table 1 summarizes the atomic quantum numbers and their allowed values.

Conceptual Questions

Exercise 1

Define the quantum numbers n, l,ml, sn, l,ml, s, and msms size 12{m rSub { size 8{s} } } {}.

Exercise 2

For a given value of nn size 12{n} {}, what are the allowed values of ll size 12{l} {}?

Exercise 3

For a given value of ll size 12{l} {}, what are the allowed values of mlml size 12{m rSub { size 8{l} } } {}? What are the allowed values of mlml size 12{m rSub { size 8{l} } } {} for a given value of nn size 12{n} {}? Give an example in each case.

Exercise 4

List all the possible values of ss size 12{s} {} and msms size 12{m rSub { size 8{s} } } {} for an electron. Are there particles for which these values are different? The same?

Problem Exercises

Exercise 1

If an atom has an electron in the n=5n=5 size 12{n=5} {} state with ml=3ml=3 size 12{m rSub { size 8{l} } =3} {}, what are the possible values of ll size 12{l} {}?

Solution

l=4, 3l=4, 3 are possible since l<nl<n size 12{l<n} {} and mllmll size 12{ lline m rSub { size 8{l} } rline {underline {<}} l} {}.

Exercise 2

An atom has an electron with ml=2ml=2 size 12{m rSub { size 8{l} } =2} {}. What is the smallest value of nn size 12{n} {} for this electron?

Exercise 3

What are the possible values of mlml size 12{m rSub { size 8{l} } } {} for an electron in the n=4n=4 size 12{n=4} {} state?

Solution

n=4l=3, 2, 1, 0ml=±3,±2, ±1, 0n=4l=3, 2, 1, 0ml=±3,±2, ±1, 0 are possible.

Exercise 4

What, if any, constraints does a value of ml=1ml=1 size 12{m rSub { size 8{l} } =1} {} place on the other quantum numbers for an electron in an atom?

Exercise 5

(a) Calculate the magnitude of the angular momentum for an l=1l=1 size 12{l=1} {} electron. (b) Compare your answer to the value Bohr proposed for the n=1n=1 size 12{n=1} {} state.

Solution

(a) 1.49×1034 Js1.49×1034 Js size 12{1 "." "49" times "10" rSup { size 8{ - "34"} } " J" cdot s} {}

(b) 1.06×1034 Js1.06×1034 Js size 12{1 "." "06" times "10" rSup { size 8{ - "34"} } " J" cdot s} {}

Exercise 6

(a) What is the magnitude of the angular momentum for an l=1l=1 size 12{l=1} {} electron? (b) Calculate the magnitude of the electron’s spin angular momentum. (c) What is the ratio of these angular momenta?

Exercise 7

Repeat Exercise 6 for l=3l=3 size 12{l=3} {}.

Solution

(a) 3.66×1034 Js3.66×1034 Js size 12{3 "." "66" times "10" rSup { size 8{ - "34"} } " J" cdot s} {}

(b) s=9.13×1035 Jss=9.13×1035 Js size 12{s=9 "." "14" times "10" rSup { size 8{ - "35"} } " J" cdot s} {}

(c) LS=123/4=4LS=123/4=4 size 12{ { {L} over {S} } = { { sqrt {"12"} } over { sqrt {3/4} } } =4} {}

Exercise 8

(a) How many angles can LL size 12{L} {} make with the zz size 12{z} {}-axis for an l=2l=2 size 12{l=2} {} electron? (b) Calculate the value of the smallest angle.

Exercise 9

What angles can the spin SS size 12{S} {} of an electron make with the zz size 12{z} {}-axis?

Solution

θ = 54.7º, 125.3º θ = 54.7º, 125.3º

Footnotes

  1. The spin quantum number s is usually not stated, since it is always 1/2 for electrons

Glossary

quantum numbers:
the values of quantized entities, such as energy and angular momentum
angular momentum quantum number:
a quantum number associated with the angular momentum of electrons
spin quantum number:
the quantum number that parameterizes the intrinsic angular momentum (or spin angular momentum, or simply spin) of a given particle
spin projection quantum number:
quantum number that can be used to calculate the intrinsic electron angular momentum along the z z -axis
z-component of spin angular momentum:
component of intrinsic electron spin along the z z -axis
magnitude of the intrinsic (internal) spin angular momentum:
given by S=ss+1hS=ss+1h
z-component of the angular momentum:
component of orbital angular momentum of electron along the z z -axis

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks