Talk:Bloch's theorem

Regarding capitalization

Stevenj: I think 134.99.136.28 was correct in capitalizing the nouns in the title of Bloch's paper: "Über die Quantenmechanik der Elektronen in Kristallgittern". Nouns are written with capital letters in German. O. Prytz 13:21, 27 December 2005 (UTC)

Ah, I didn't realize that. —Steven G. Johnson 19:38, 30 December 2005 (UTC)

Tight binding

I've learned that there are s,p and d bloch waves in transition metals. And that the rules for dipole radiation are very similar to free atoms with s,p,d orbitals. How is the connection? Tight binding model? Would this improve this article? Arnero 20:24, 26 January 2007 (UTC)

Tight-binding approximations are just a simple approximate expansion basis for the Bloch eigenfunctions in electronic systems. It doesn't have anything conceptually to do with Bloch's theorem per se. For example, you could just as easily use a tight-binding approximation in a non-periodic structure where Bloch's theorem does not apply. —Steven G. Johnson 05:05, 27 January 2007 (UTC)

about F.B.Z.

I suggest that we should note some mathematics descriptions about the first Brillouin zone in an article.

Epsilon?

Shouln't the Greek letter epsilon (ε) be used instead of "Є"?

The derivation

The derivation of the Bloch theorem is added. I am a new wikipedian and not very proficient in the wiki markup language and the formula is not very satisfactory. Could anyone help with that? Thank youTschijnmotschau (talk) 14:12, 3 December 2010 (UTC)

Corrected 2 pi error

The original derivation failed to properly divide k by 2pi, because the normalization convention for the reciprocal lattice vectors is that a_i \dot b_j = 2 \pi \delta_{ij}. I fixed it. — Preceding unsigned comment added by 68.55.253.191 (talk) 22:55, 19 September 2013 (UTC)

The article states the convention it uses earlier in the proof ("These are three vectors b₁, b₂, b₃ (with units of inverse length), with the property that a_i · b_i = 1, but a_i · b_j = 0 when i ≠ j."). That edit introduced an error. I don't mind switching conventions, but it has to be done in the entirety of the article.Headbomb {talk / contribs / physics / books} 11:06, 20 September 2013 (UTC)

Actually the article seems to have a crysis of identity... I'll investigate tonight after I get back from work. Headbomb {talk / contribs / physics / books} 11:15, 20 September 2013 (UTC)

I agree with 68.55.253.191 that it's better to use a_i \dot b_j = 2 \pi \delta_{ij}, because the whole article is generally using angular wavevectors not normal wavevectors. I changed it here. I think everything is correct and consistent now, does everyone agree? --Steve (talk) 17:40, 20 September 2013 (UTC)

I agree physical wavevectors with a.b = 2π δ_ij is the better convention. I'm just not sure the exponentials now have the correct factors. I said I would check tonight, but I'm tired from a very long day at work. I'll check them all tomorrow to make sure. Headbomb {talk / contribs / physics / books} 01:07, 21 September 2013 (UTC)

Definition of "Bloch wave"

I just reverted a change. The definition used throughout the article is:

A Bloch state is any state that is both (1) an energy eigenstate, and (2) can be written as $\psi (\mathbf {r} )=\mathrm {e} ^{\mathrm {i} \mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} )$ where u has the same periodicity as the atomic structure of the crystal.

User:Edib76 seems to disagree with this definition, preferring to use the alternative definition

A Bloch state is any state that can be written as $\psi (\mathbf {r} )=\mathrm {e} ^{\mathrm {i} \mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} )$ where u has the same periodicity as the atomic structure of the crystal.

We should use whatever definition is more common, according to popular textbooks and other sources. I believe it's the first one. But maybe I'm remembering wrong!! I don't have any books with me right this second. Let's work together to figure out which definition is more standard!

If the second one is actually the standard definition (which I don't think it is, but again I could be wrong), then we need to change it consistently throughout the whole article, not just in one section. That's why I have reverted User:Edib76's change for now, pending this investigation. :-D Thanks in advance!! --Steve (talk) 01:49, 13 January 2014 (UTC)

UPDATE: Here are some references.

• Kittel Intro to Solid-State Physics page 259:

"A function of the form (19) [i.e. $\psi _{k}(r)=e^{ik\cdot r}u_{k}(r)$] is known as a Bloch function"

which is the second definition above.

• Kittel Quantum Theory of Solids page 180:

"The function $u_{k}(x)$ of the Bloch function $\phi _{k}(x)=e^{ikx}u_{k}(x)$ satisfies the equation $({\frac {1}{2m}}(p+k)^{2}+V(x))u_{k}(x)=\epsilon _{k}u_{k}(x)$

which suggests the first definition above, i.e. it suggests that anything called a "Bloch function" must be a stationary state.

• Ashcroft and Mermin do not use the terms "Bloch wave" or "Bloch function" or "Bloch wavefunction" in the main discussion of Bloch's theorem (chapter 8) as far as I see. They do use the term "Bloch electron" to mean "an electron that "obeys a one-particle Schrodinger equation with a periodic potential", whether or not that electron is in a stationary state (they talk about "stationary states of Bloch electrons". Later in the book (page 185 and 215) they mention "Bloch wave" and "Bloch wave function" but in a way where it's ambiguous whether or not the term only refers to stationary states. Oh, and they use the term "Bloch form" as synonymous with $e^{ikx}u(x)$ (page 139). So they don't strongly endorse either definition, but maybe there is an implicit slight preference for the second definition.

• Griffiths (3rd edition) describes Bloch's theorem but doesn't use the terms "Bloch wave" or "Bloch function" or "Bloch wavefunction".

• Yu and Cardona (page 20) use "Bloch function" to refer to the "form" (i.e. a plane-wave times a periodic function), which suggests the second definition above.

It seems to me, we should use the second definition: A Bloch wave is any wave of the form $e^{ikx}u(x)$, whether or not it is an energy eigenstate. I will change the article ... --Steve (talk) 19:14, 14 January 2014 (UTC)

@Footlessmouse: I appreciate the helpful edits you've made such as these [1], but the definition in the lead in not supported by the source. The definition I've gone back to reflects the survey of the literature done by the guy above. In your edit summary, you seem to be saying a Bloch wave and a Bloch wavefunction are different things? Kittel treats them as the same: "A one-electron wavefunction of the form (7) is called a Bloch function..." (Introduciton To Solid State Physics, 8th edition, pp.167)

My best guess is that you're trying to lay out a rigorous distinction between an abstract state in Hilbert space and its representation in position space, but that is far too technical for this topic. We should write one level down. But maybe you mean something else? It would be helpful if you could point out some sources that support this distinction. $\langle$ Forbes₇₂ | Talk $\rangle$ 00:42, 27 August 2020 (UTC)

@Forbes₇₂. Yes, I guess you could say I was trying to lay out a distinction between an abstract state in Hilbert space and its representation in position space, though from a less formal point of view, you could just say that a wave is a physical phenomenon, while a wave-function is any mathematical description of the wave. A wave obeys a wave-function.

From Wave: "In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities, sometimes as described by a wave equation."

From Wave function: "A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system."

As you can see from the survey above, none of the authors of these texts support referring to the wave and wave-function as the same thing. They all are giving definition of the wave function by its mathematical form. The Bloch electron in Ashcroft and Mermin is simply an electron that obeys this wave-function, all others describe only form. Kittel only refers to the wavefunction being a Bloch wave function, not the Bloch wave being a Bloch wave-function. As you will find, no literature will ever support calling a wave and wave function the same thing, as it is mathematically incoherent. I will not revert your reversion, but I was not wrong for fixing the ambiguity. Footlessmouse (talk) 02:19, 27 August 2020 (UTC)

For the record, I agree you are correct on the technical point, even if I don't think such a fine distinction is useful for a basic condensed matter physics article. But it doesn't matter what I think, or even what is true. Wikipedia is not designed to be correct, it's designed to be verifiable. Wikipedia just copies whatever the textbooks/papers/etc say. You mention a couple articles, but Wikipedia itself is not an acceptable source, and you can't combine two points together like that anyway. In order to discuss the distinction on Wikipedia, we need to find a source that explicitly discusses such a distinction in this context. Apologies if the policy seems strange; the standards for writing a Wikipedia article are surprisingly different than those for writing academic literature. $\langle$ Forbes₇₂ | Talk $\rangle$ 03:37, 27 August 2020 (UTC)

(Reply to a now deleted comment)

It's a good point that "Bloch wave" is actually fairly rare terminology. The most common/general terminology seems to be "Bloch's theorem": I can confirm it appears explicitly in Kittel Solid state, Ashcroft/Mermin, Griffiths QM, as cited above, and in several other books. As you mention, that does mean rewriting the lead. So let's talk about it. How about something like:

"In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. These solutions, sometimes known as Bloch functions, are eigenstates in energy, and serve as a suitable basis for the wave functions of electrons in crystalline solids. Mathematically, they are written:

where $\mathbf {r}$ is position, $\psi$ is the Bloch wave, $u$ is a periodic function with the same periodicity as the crystal, the wave vector $\mathbf {k}$ is the crystal momentum vector, $\mathrm {e}$ is Euler's number, and $\mathrm {i}$ is the imaginary unit.

The description of electrons in terms of Bloch functions underlies the concept of electronic band structures.

These eigenstates are written with subscripts as $\psi _{n\mathbf {k} }$, where $n$ is a discrete index, called the band index, which is present because there are many different wavefunctions with the same $\mathbf {k}$ (each has a different periodic component $u$). Within a band (i.e., for fixed $n$), $\psi _{n\mathbf {k} }$ varies continuously with $\mathbf {k}$, as does its energy. Also, $\psi _{n\mathbf {k} }$, is unique only up to a constant reciprocal lattice vector $\mathbf {K}$, or, $\psi _{n\mathbf {k} }=\psi _{n(\mathbf {k+K} )}$. Therefore, the wave vector $\mathbf {k}$ can be restricted to the first Brillouin zone of the reciprocal lattice without loss of generality."

This follows loosely follows Kittel, without defining "Bloch wavefunction" since the term doesn't appear there. It still could be improved, but I wonder what you think of this. $\langle$ Forbes₇₂ | Talk $\rangle$ 07:33, 28 August 2020 (UTC)

Hi @Forbes₇₂, sorry, I didn't realize you were going to respond. If I may take the moment, I am sorry for the previous comments. I was actually mistaken and feel pretty ridiculous about it. (For some reason I started to confuse wave-function and wave-equation. I guess the wave=wavefunction didn't sit right and I was otherwise just trying to justify it) I agree, though, that "Bloch wave" is seldom used and it might be better to use "Bloch theorem". Your proposed intro has my blessing, though I would probably want to work "Bloch electron" in there. The first sentence is spot on, I don't think we could do much better. You can also say the Bloch function is also called "Bloch state" and "Bloch wavefunction", but I don't think they have to be energy eigenstates in general, the literature is ambiguous on that, but they might use different terms. Otherwise we'll have to decide on how much information is really necessary, as you can always say that a state of that form can be written as a sum of energy eignestates, as they represent a complete basis set. Footlessmouse (talk) 08:25, 28 August 2020 (UTC)

No problem. It was a little confusing to understand what you meant, but you brought up some good ideas. Those kinds of tweaks sound reasonable enough. I've formally proposed the move in a new section below. Can you weigh in there? (see these discussions for examples: [2] or [3]) If there's no objection we can request a technical move in a week or so. $\langle$ Forbes₇₂ | Talk $\rangle$ 22:30, 28 August 2020 (UTC)

Hey @Forbes72, should we replace the introduction with the one you prepared? No rush, but I wanted to ask in case you forgot about it. Footlessmouse (talk) 07:36, 11 September 2020 (UTC)

 Done Thanks for the heads up. I still think eventually the article should probably move "Theorem" to "theorem", but for now I'm more interested on working on some other articles. Feel free to make your own improvements. I keep an eye on this article, but I'm not here to WP:OWN. $\langle$ Forbes₇₂ | Talk $\rangle$ 22:00, 11 September 2020 (UTC)

Dear all, I think there still is some confusion in the introduction. Bloch theory and Bloch Waves are two separate (but related) concepts. The Bloch theorem states that all eigenfunctions in a periodic lattice are of the form of a plane wave times a periodic function. The wave vector of these plane waves is the crystal momentum of the eigen state. A Bloch wave is any wave that can be written on this form, not necessarily also an eigen state. (This definition is in agreement with both Kitel and A&M, although Kitel on page 180 might lead to confusion). Bloch Waves are used to calculate the eigen states in a solid. One starts with a set of atom centred functions. From those one can generate periodic functions by repeating the local function in each unit cell. Multiply with a plane wave generates a Bloch wave. Linear combinations of the Bloch Waves are Eigen-states in the periodic potential. Otherwise thanks for the good info! Maurits W. Haverkort (talk) 21:14, 23 November 2020 (UTC)

Hi Maurits W. Haverkort! Please note, as is noted above, neither Kittel nor A&M nor any other standard CM textbook mentions "Bloch waves", but rather Bloch states and functions and electrons, etc. There is actually more confusion than that. I started a literature survey to try to pin down what they say, but some of the definitions are contradictory. Especially, for instance, that they don't have to be eigenstates - that depends on what book you're reading. The best route forward, that I see, is adding a "definition" section that tries to lay all that info out. Otherwise, I am a little confused about your statement because the introduction does not say that Bloch's theorem is the same thing as Bloch states. In fact, it defines Bloch's theorem as stating that the solutions of Schrodinger's equation in a crystal are given by Bloch states, which is supported by the literature (it does say that Bloch electrons are also called Bloch waves, which is not true for many reasons). Are you trying to argue they should have separate pages? If so, I don't think we have enough information to fill out a second article, and if we did we would have to decide on a title supported by the references, which is difficult as they all use different terms for some of the same concepts and they use the same terms for other concepts in which they all slightly differ. That is a primary reason the page is now titled "Bloch's theorem", as it is unambiguous, used in all the textbooks, and has the same definition everywhere (not to mention it is manifestly more notable, being mentioned every time any of the other terms are used). Let me know if I have misunderstood anything or if you have any other questions. Footlessmouse (talk) 21:59, 23 November 2020 (UTC)

Dear Footlessmouse! thanks for the comment. I indeed did not make the distinction between waves, wave functions, and states in the framework of quantum mechanics. I know there is a difference, but I do not think it is important for the current discussion. The question I want to discuss is if a Bloch wave-function is necessarily an eigen wave-function of the Hamiltonian, or if it can be more general defined. (Equivalent, is a Bloch state necessarily and eigen state of the Hamiltonian). The second sentence of the article now states that they have to be equivalent. Looking at the citations from literature above it is not used by these authors in quite that strict form. (Kitel contradict himself, in two different books A&M evade the pure definition by talking about Bloch electrons and Bloch form for the states or wave functions they encounter, Yu and Cardona state it does not need to be an eigen function.) Note that we all agree that eigen functions in a periodic solid are Bloch functions, but not all authors would state that a Bloch function is necessarily an eigenfunction of a periodic Hamiltonian.

The reason for using Bloch functions as not being eigen functions is that if one starts from localised orbitals (atomic like or Wannier functions) and wants to create eigen states of the periodic lattice one in an intermediate step encounters functions that have the form of a periodic function multiplied with a plane wave, but are not eigen functions yet. These functions "serve as a suitable basis for the wave functions of electrons in crystalline solids." as they decouple wave functions with different crystal momentum. The statement that the eigen functions form a suitable basis as now made in the second sentence of this article is true but trivial.

I would suggest a change of the form:

In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. Mathematically, they are written:

Bloch function

$\psi (\mathbf {r} )=\mathrm {e} ^{\mathrm {i} \mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} )$

where $\mathbf {r}$ is position, $\psi$ is the wave function, $u$ is a periodic function with the same periodicity as the crystal, the wave vector $\mathbf {k}$ is the crystal momentum vector, $\mathrm {e}$ is Euler's number, and $\mathrm {i}$ is the imaginary unit.

Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable basis for the wave functions (states) of electrons in crystalline solids. Some but not all authors --citations Kitel 1+2 A&M Yu and Cardona-- require Bloch functions to be additional eigen functions of the Hamiltonian. Maurits W. Haverkort (talk) 07:33, 24 November 2020 (UTC)

I see what you mean. I don't think anyone will object if you made that change, it sort of avoids the whole dispute. Like I said, different textbooks treat them differently, especially "Bloch states", as you can imagine, a Bloch state is almost always going to be an eigenstate as there is not much point talking about it otherwise; the Bloch function, Bloch wavefunction, and "Bloch form" are all ambiguous. I think a definitions section will help clear up any other misunderstandings. Footlessmouse (talk) 08:19, 24 November 2020 (UTC)

Maurits W. Haverkort Edit: I think the last sentence can be left off for the time being, and if we create the definitions section then a short summary of that section can go there with no need to double cite, it might could be read as OR the way it is now, I'm not sure. The content prior to that implies that it is more general and I think the details can be relegated to the body. Footlessmouse (talk) 08:25, 24 November 2020 (UTC)

Missing Definitions For The Formula and etc.

e, i and r are not defined.
The names of the Greek letters are missing too.
They are not linked to the Greek Alphabet.
botwork
— Preceding unsigned comment added by Jangirke (talk • contribs) 22:41, 19 February 2014 (UTC)

I added e and i at the top. I don't see any reason to link the greek letters. The fact that ψ is called "psi" not "funny-looking greek letter" is a good thing to know but it's off-topic when discussing Bloch waves. (Just my opinion.) --Steve (talk) 03:33, 20 February 2014 (UTC)

Requested move 28 August 2020

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: page moved. (non-admin closure) ~SS49~ {talk} 00:21, 5 September 2020 (UTC)

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Bloch wave → Bloch's Theorem – Bloch's theorem is a much more widely used name. Standard reference books such as David Griffith's Introduction to Quantum Mechanics, Ashcroft and Mermin's Solid State Physics, and Charles Kittel's Introduction to Solid State Physics all list "Bloch's theorem" in their text and also list it in their back indices, whereas none of them use the term "Bloch wave" at all. It's also much less ambiguous on the precise meaning. (Do Bloch waves mean only eigenstates or also combinations of eigenstates?) $\langle$ Forbes₇₂ | Talk $\rangle$ 22:00, 28 August 2020 (UTC)

• Support per nominator. Bloch's theorem is regularly discussed in the literature and is an appropriate name for an overview of the topic. Footlessmouse (talk) 23:38, 28 August 2020 (UTC)
• Support. Further reference Sakurai's Modern Quantum Mechanics — Preceding unsigned comment added by 2600:6C58:4B80:2727:6463:9C11:4331:9178 (talk) 02:29, 31 August 2020 (UTC)

Technical points: In the proposal template, I've mistakenly written "Bloch's Theorem" instead of "Bloch's theorem". Per the Wikipedia standard style, it should be the latter. The disambiguation with Bloch's theorem (complex variables) can be accomplished by a hatnote,

This article is about quantum mechanics. For the theorem of complex analysis, see Bloch's theorem (complex variables).

since the theorem concerning crystals is roughly ten times as popular [4] as the theorem concerning holomorphic functions.$\langle$ Forbes₇₂ | Talk $\rangle$ 21:14, 31 August 2020 (UTC)

References

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The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Small mistake : wrong image in the "Applications and consequences" section

Dear all,

It seems to me that the last pair of images, in the figure comparing two expressions of the same Bloch state, shows two identical decompositions. The same image must have been taken twice by mistake !

LaurentV (talk) 08:29, 2 October 2020 (UTC)

Dataset of the image

Interesting I think you are right, I left a message to Sbyrnes321 to check if he is willing either to fix the image or to share the dataset to rebuild the graph. comment added by flyredeagle

The image detail page File:BlochWaves1D.svg has source code, but I'm very confused what you think the problem is. Are you referring to the fact that the bottom-left plot and bottom-right plot are the same curves? There's supposed to be the same curves. That's the whole point. Or are you talking about something else? Can you explain more specifically? Thanks. --Steve (talk) 13:35, 27 October 2020 (UTC)

Please correct me if I'm wrong, but both users were referring to the use of the same plot in the bottom-left and bottom-right of that figure. Reinforcing what was said by Steve, they are supposed to be the same, there is no mistake. Footlessmouse (talk) 16:42, 27 October 2020 (UTC)

Bloch waves neq Bloch electrons

Hi all, I will spend some time looking over sources to see what we can do about enhancing this article. I do, however want to point out Bloch electrons are not sometimes called Bloch waves. I have removed the vast majority of the backlinks to Bloch wave (most of which were really to Bloch wave-function, but "function" was left outside the link). If it is to be insisted that we keep it, it should go with the other names as it is clearly not a synonym for Bloch electron. Thanks! Footlessmouse (talk) 05:23, 28 October 2020 (UTC)

I think the article splits the definition of Bloch's waves (which are always called Bloch's functions on purpose to avoid confusion and mention once to "less often called Bloch's waves") referenced in the first part from bloch electrons (i.e. electrons in a band - e.g. see Ashcroft topic index pp==800 which refers to a few like: density of states pp=143-146, dynamics pp=214-241, holes pp=225-229) referenced in the second part in a decent neat way, what is the distinction between an electron and a wave in any case ? As discussed in august Bloch waves is more of an historical term and is best off to be buried in a corner. The major prob of the discussion in august to me is the fact there was no Bloch theorem page which instead is the best recognizable term in recent years.

Flyredeagle (talk) 07:43, 24 December 2020 (UTC)

Okay, so I found a reliable source that talks about Bloch waves. When I first made the proposal above, I was new to Wikipedia, I've since gotten much better at looking for sources. This source, though is not a standard CM textbook, states that "The solutions to the Schrodinger equation which always have the required translation property are known as Bloch waves. Since these wave functions..." It appears to hold Bloch waves and Bloch wave functions as synonymous (this is reinforced later when it says "The main point to remember is that each Bloch wave is associated with just one k(j) but it is a continuously varying function of r").

Maybe some improvement on the text of the article can help to clarify that Bloch's wave functions are actually eigenstates i.e. with just one k(j), eigenstates are mentioned a few times but maybe the text can be improved. Flyredeagle (talk) 07:43, 24 December 2020 (UTC)

However, in a very unfortunate choice of terminology, it specifically defines the Bloch's function as the potential part alone, not the entire wavefunction including the periodic potential and plane wave. I believe this calls for a further, more detailed survey of the literature. Footlessmouse (talk) 21:30, 28 October 2020 (UTC)

References

In terms of references for the definition of Bloch Waves I would not use a specialist text book but a major undergrad / grad book, therefore missing a definition of Bloch's waves in a major such book I used instead the definition of the opposite term "Bloch Electrons" as you see in Ashcroft.

I in fact disagree on the last distinction "potential part alone" in your book: an electron without kinetic energy does not make much sense, but some calls Bloch wave functions stationary waves and therefore they consider them with "total" and "average" kinetic energy zero, that is where probably the term "potential part alone" comes from. In some texts they call them "electrons at rest" in a crystal which, although it may be "suggestive" of some magic that electrons can do, is even more confusing and is a plain heuristic narrative, this in fact refers to the fact that the stationary points of the wave don't actually move.

Flyredeagle (talk) 07:43, 24 December 2020 (UTC)

First Thanks for the comments, I think we shall promote any genuine contributor. Then for text improvements please go ahead, for major content improvements please add here in this talk a requirement section / requested content and then we can discuss it further or just type in here some proposed extra content and then we promote it to the page. (e.g. maybe it goes in other articles). In regards to literature search I would close off with this "Bloch Wave definition" discussion and move on to more interesting topics (i.e. not just definitions), e.g. spend effort on adding new content.

Flyredeagle (talk) 07:43, 24 December 2020 (UTC)

The first graph is wrong?

The graph with the caption,

"Solid line: A schematic of the real part of a typical Bloch state in one dimension. The dotted line is from the eik·r factor. The light circles represent atoms."

is not right. 98.115.95.31 (talk) 17:22, 25 August 2022 (UTC)